Average Error: 20.0 → 9.7
Time: 15.4s
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 r324933 = x;
        double r324934 = y;
        double r324935 = r324933 * r324934;
        double r324936 = r324933 + r324934;
        double r324937 = r324936 * r324936;
        double r324938 = 1.0;
        double r324939 = r324936 + r324938;
        double r324940 = r324937 * r324939;
        double r324941 = r324935 / r324940;
        return r324941;
}
↓
double f(double x, double y) {
        double r324942 = x;
        double r324943 = cbrt(r324942);
        double r324944 = r324943 * r324943;
        double r324945 = r324942 * r324942;
        double r324946 = y;
        double r324947 = 2.0;
        double r324948 = r324947 * r324942;
        double r324949 = r324948 + r324946;
        double r324950 = r324946 * r324949;
        double r324951 = r324945 + r324950;
        double r324952 = r324951 / r324943;
        double r324953 = r324944 / r324952;
        double r324954 = r324942 + r324946;
        double r324955 = 1.0;
        double r324956 = r324954 + r324955;
        double r324957 = r324946 / r324956;
        double r324958 = r324953 * r324957;
        return r324958;
}

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))))