Average Error: 19.9 → 0.4
Time: 5.6s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
↓
\[\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ x y)) (/ (+ x y) (/ y (+ x (+ y 1.0))))))
double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}
↓
double code(double x, double y) {
	return (x / (x + y)) / ((x + y) / (y / (x + (y + 1.0))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.2
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation

  1. Initial program 19.9

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

  2. Applied associate-/l*_binary6411.8

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

  3. Simplified9.7

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

  4. Applied *-un-lft-identity_binary649.7

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

  5. Applied *-un-lft-identity_binary649.7

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

  6. Applied times-frac_binary649.7

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

  7. Applied times-frac_binary646.0

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

  8. Applied associate-/r*_binary640.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021313 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))