Average Error: 20.0 → 9.7
Time: 16.6s
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 r287316 = x;
        double r287317 = y;
        double r287318 = r287316 * r287317;
        double r287319 = r287316 + r287317;
        double r287320 = r287319 * r287319;
        double r287321 = 1.0;
        double r287322 = r287319 + r287321;
        double r287323 = r287320 * r287322;
        double r287324 = r287318 / r287323;
        return r287324;
}
↓
double f(double x, double y) {
        double r287325 = x;
        double r287326 = cbrt(r287325);
        double r287327 = r287326 * r287326;
        double r287328 = r287325 * r287325;
        double r287329 = y;
        double r287330 = 2.0;
        double r287331 = r287330 * r287325;
        double r287332 = r287331 + r287329;
        double r287333 = r287329 * r287332;
        double r287334 = r287328 + r287333;
        double r287335 = r287334 / r287326;
        double r287336 = r287327 / r287335;
        double r287337 = r287325 + r287329;
        double r287338 = 1.0;
        double r287339 = r287337 + r287338;
        double r287340 = r287329 / r287339;
        double r287341 = r287336 * r287340;
        return r287341;
}

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))))