Animation: Fix case where QEasingCurve::valueForProgress returns nan

Previously, we would divide by zero in BezierEase::findTForX if factorT3
was zero when solving the cubic equation.

This change fixes the problem by adding solutions for the special cases
where the cubic equation can be reduced to a quadratic or linear
equation.

This change also adds tests that cover cases where the equation becomes
quadratic, linear or invalid.

Task-number: QTBUG-67061
Change-Id: I2b59f7e0392eb807663c3c8927509fd8b226ebc7
Reviewed-by: Christian Stromme <christian.stromme@qt.io>

2 files changed, 124 insertions, 21 deletions

diff --git a/src/corelib/tools/qeasingcurve.cpp b/src/corelib/tools/qeasingcurve.cpp
index 0b8fa4ca74..e66db58ed7 100644
--- a/src/corelib/tools/qeasingcurve.cpp
+++ b/src/corelib/tools/qeasingcurve.cpp
@@ -797,27 +797,60 @@ struct BezierEase : public QEasingCurveFunction
         return t3;
     }
 
-     qreal static inline findTForX(const SingleCubicBezier &singleCubicBezier, qreal x)
-     {
-         const qreal p0 = singleCubicBezier.p0x;
-         const qreal p1 = singleCubicBezier.p1x;
-         const qreal p2 = singleCubicBezier.p2x;
-         const qreal p3 = singleCubicBezier.p3x;
-
-         const qreal factorT3 = p3 - p0 + 3 * p1 - 3 * p2;
-         const qreal factorT2 = 3 * p0 - 6 * p1 + 3 * p2;
-         const qreal factorT1 = -3 * p0 + 3 * p1;
-         const qreal factorT0 = p0 - x;
-
-         const qreal a = factorT2 / factorT3;
-         const qreal b = factorT1 / factorT3;
-         const qreal c = factorT0 / factorT3;
-
-         return singleRealSolutionForCubic(a, b, c);
-
-         //one new iteration to increase numeric stability
-         //return newtonIteration(singleCubicBezier, t, x);
-     }
+    bool static inline almostZero(qreal value)
+    {
+        // 1e-3 might seem excessively fuzzy, but any smaller value will make the
+        // factors a, b, and c large enough to knock out the cubic solver.
+        return value > -1e-3 && value < 1e-3;
+    }
+
+    qreal static inline findTForX(const SingleCubicBezier &singleCubicBezier, qreal x)
+    {
+        const qreal p0 = singleCubicBezier.p0x;
+        const qreal p1 = singleCubicBezier.p1x;
+        const qreal p2 = singleCubicBezier.p2x;
+        const qreal p3 = singleCubicBezier.p3x;
+
+        const qreal factorT3 = p3 - p0 + 3 * p1 - 3 * p2;
+        const qreal factorT2 = 3 * p0 - 6 * p1 + 3 * p2;
+        const qreal factorT1 = -3 * p0 + 3 * p1;
+        const qreal factorT0 = p0 - x;
+
+        // Cases for quadratic, linear and invalid equations
+        if (almostZero(factorT3)) {
+            if (almostZero(factorT2)) {
+                if (almostZero(factorT1))
+                    return 0.0;
+
+                return -factorT0 / factorT1;
+            }
+            const qreal discriminant = factorT1 * factorT1 - 4.0 * factorT2 * factorT0;
+            if (discriminant < 0.0)
+                return 0.0;
+
+            if (discriminant == 0.0)
+                return -factorT1 / (2.0 * factorT2);
+
+            const qreal solution1 = (-factorT1 + std::sqrt(discriminant)) / (2.0 * factorT2);
+            if (solution1 >= 0.0 && solution1 <= 1.0)
+                return solution1;
+
+            const qreal solution2 = (-factorT1 - std::sqrt(discriminant)) / (2.0 * factorT2);
+            if (solution2 >= 0.0 && solution2 <= 1.0)
+                return solution2;
+
+            return 0.0;
+        }
+
+        const qreal a = factorT2 / factorT3;
+        const qreal b = factorT1 / factorT3;
+        const qreal c = factorT0 / factorT3;
+
+        return singleRealSolutionForCubic(a, b, c);
+
+        //one new iteration to increase numeric stability
+        //return newtonIteration(singleCubicBezier, t, x);
+    }
 };
 
 struct TCBEase : public BezierEase
diff --git a/tests/auto/corelib/tools/qeasingcurve/tst_qeasingcurve.cpp b/tests/auto/corelib/tools/qeasingcurve/tst_qeasingcurve.cpp
index 0714883855..79309f960d 100644
--- a/tests/auto/corelib/tools/qeasingcurve/tst_qeasingcurve.cpp
+++ b/tests/auto/corelib/tools/qeasingcurve/tst_qeasingcurve.cpp
@@ -54,6 +54,7 @@ private slots:
     void testCbrtDouble();
     void testCbrtFloat();
     void cpp11();
+    void quadraticEquation();
 };
 
 void tst_QEasingCurve::type()
@@ -804,5 +805,74 @@ void tst_QEasingCurve::cpp11()
 #endif
 }
 
+void tst_QEasingCurve::quadraticEquation() {
+    // We find the value for a given time by solving a cubic equation.
+    //     ax^3 + bx^2 + cx + d = 0
+    // However, the solver also needs to take care of cases where a = 0,
+    // b = 0 or c = 0, and the equation becomes quadratic, linear or invalid.
+    // A naive cubic solver might divide by zero and return nan, even
+    // when the solution is a real number.
+    // This test should triggers those cases.
+
+    {
+        // If the control points are spaced 1/3 apart of the distance of the
+        // start- and endpoint, the equation becomes linear.
+        QEasingCurve test(QEasingCurve::BezierSpline);
+        const qreal p1 = 1.0 / 3.0;
+        const qreal p2 = 1.0 - 1.0 / 3.0;
+        const qreal p3 = 1.0;
+
+        test.addCubicBezierSegment(QPointF(p1, 0.0), QPointF(p2, 1.0), QPointF(p3, 1.0));
+        QVERIFY(qAbs(test.valueForProgress(0.25) - 0.15625) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.5) - 0.5) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.75) - 0.84375) < 1e-6);
+    }
+
+    {
+        // If both the start point and the first control point
+        // are placed a 0.0, and the second control point is
+        // placed at 1/3, we get a case where a = 0 and b != 0
+        // i.e. a quadratic equation.
+        QEasingCurve test(QEasingCurve::BezierSpline);
+        const qreal p1 = 0.0;
+        const qreal p2 = 1.0 / 3.0;
+        const qreal p3 = 1.0;
+        test.addCubicBezierSegment(QPointF(p1, 0.0), QPointF(p2, 1.0), QPointF(p3, 1.0));
+        QVERIFY(qAbs(test.valueForProgress(0.25) - 0.5) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.5) - 0.792893) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.75) - 0.950962) < 1e-6);
+    }
+
+    {
+        // If both the start point and the first control point
+        // are placed a 0.0, and the second control point is
+        // placed close to 1/3, we get a case where a = ~0 and b != 0.
+        // It's not truly a quadratic equation, but should be treated
+        // as one, because it causes some cubic solvers to fail.
+        QEasingCurve test(QEasingCurve::BezierSpline);
+        const qreal p1 = 0.0;
+        const qreal p2 = 1.0 / 3.0 + 1e-6;
+        const qreal p3 = 1.0;
+        test.addCubicBezierSegment(QPointF(p1, 0.0), QPointF(p2, 1.0), QPointF(p3, 1.0));
+        QVERIFY(qAbs(test.valueForProgress(0.25) - 0.499999) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.5) - 0.792892) < 1e-6);
+        QVERIFY(qAbs(test.valueForProgress(0.75) - 0.950961) < 1e-6);
+    }
+
+    {
+        // A bad case, where the segment is of zero length.
+        // However, it might still happen in user code,
+        // and we should return a sensible answer.
+        QEasingCurve test(QEasingCurve::BezierSpline);
+        const qreal p0 = 0.0;
+        const qreal p1 = p0;
+        const qreal p2 = p0;
+        const qreal p3 = p0;
+        test.addCubicBezierSegment(QPointF(p1, 0.0), QPointF(p2, 1.0), QPointF(p3, 1.0));
+        test.addCubicBezierSegment(QPointF(p3, 1.0), QPointF(1.0, 1.0), QPointF(1.0, 1.0));
+        QCOMPARE(test.valueForProgress(0.0), 0.0);
+    }
+}
+
 QTEST_MAIN(tst_QEasingCurve)
 #include "tst_qeasingcurve.moc"