Average Error: 19.6 → 0.1
Time: 5.4s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
↓
\[\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x} \]
\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{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}

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

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation

  1. Initial program 19.6

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

  2. Simplified19.6

    \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

  3. Applied *-un-lft-identity_binary6419.6

    \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot \mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

  4. Applied times-frac_binary6411.5

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

  5. Simplified11.5

    \[\leadsto \color{blue}{x} \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

  6. Simplified23.8

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

  7. Applied unpow2_binary6423.8

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

  8. Applied unpow3_binary6423.8

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

  9. Applied distribute-rgt-out_binary6411.5

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

  10. Applied *-un-lft-identity_binary6411.5

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

  11. Applied times-frac_binary646.0

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

  12. Applied associate-*r*_binary644.2

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

  13. Simplified4.2

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

  14. Applied associate-*l/_binary644.2

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2022076 
(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))))