Average Error: 20.0 → 9.7
Time: 17.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r277108 = x;
        double r277109 = y;
        double r277110 = r277108 * r277109;
        double r277111 = r277108 + r277109;
        double r277112 = r277111 * r277111;
        double r277113 = 1.0;
        double r277114 = r277111 + r277113;
        double r277115 = r277112 * r277114;
        double r277116 = r277110 / r277115;
        return r277116;
}
↓
double f(double x, double y) {
        double r277117 = x;
        double r277118 = cbrt(r277117);
        double r277119 = r277118 * r277118;
        double r277120 = r277117 * r277117;
        double r277121 = y;
        double r277122 = 2.0;
        double r277123 = r277122 * r277117;
        double r277124 = r277123 + r277121;
        double r277125 = r277121 * r277124;
        double r277126 = r277120 + r277125;
        double r277127 = r277126 / r277118;
        double r277128 = r277119 / r277127;
        double r277129 = r277117 + r277121;
        double r277130 = 1.0;
        double r277131 = r277129 + r277130;
        double r277132 = r277121 / r277131;
        double r277133 = r277128 * r277132;
        return r277133;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
Herbie9.7
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.0

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]

  2. Taylor expanded around 0 20.0

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left({x}^{2} + \left({y}^{2} + 2 \cdot \left(x \cdot y\right)\right)\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

  3. Simplified20.0

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot x + y \cdot \left(2 \cdot x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)}\]
  4. Using strategy rm

  5. Applied times-frac9.3

    \[\leadsto \color{blue}{\frac{x}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt9.7

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}\]

  8. Applied associate-/l*9.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}}} \cdot \frac{y}{\left(x + y\right) + 1}\]

  9. Final simplification9.7

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1))))