Average Error: 19.1 → 0.7
Time: 6.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{x}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]
\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}}{\frac{x + y}{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{x}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\right)
double code(double x, double y) {
	return ((double) (((double) (x * y)) / ((double) (((double) (((double) (x + y)) * ((double) (x + y)))) * ((double) (((double) (x + y)) + 1.0))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) cbrt(x)) / ((double) (((double) (x + y)) / ((double) cbrt(x)))))) * ((double) (((double) (((double) cbrt(x)) / ((double) (x + y)))) * ((double) (y / ((double) (((double) (x + y)) + 1.0))))))));
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.1
Target0.1
Herbie0.7
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.1

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

  3. Applied associate-*l*19.1

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

  4. Applied add-cube-cbrt19.4

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

  5. Applied associate-*l*19.4

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

  6. Applied times-frac8.4

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

  7. Simplified8.4

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))))