Average Error: 19.9 → 0.1
Time: 27.3s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{x + \left(y + 1\right)}}{x + y}\]
\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} \cdot \frac{y}{x + \left(y + 1\right)}}{x + y}
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (/ (* (/ x (+ x y)) (/ y (+ x (+ y 1.0)))) (+ x y)))
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)) * (y / (x + (y + 1.0)))) / (x + y);
}

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

Derivation

  1. Initial program 19.9

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

  3. Applied associate-/l*_binary64_1264011.3

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

  4. Simplified9.5

    \[\leadsto \frac{x}{\color{blue}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\frac{y}{x + \left(y + 1\right)}}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity_binary64_126959.5

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

  7. Applied *-un-lft-identity_binary64_126959.5

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

  8. Applied times-frac_binary64_127019.5

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

  9. Applied times-frac_binary64_127015.7

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

  10. Applied *-un-lft-identity_binary64_126955.7

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

  11. Applied times-frac_binary64_127010.5

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

  12. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{x + y}} \cdot \frac{x}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}\]
  13. Using strategy rm

  14. Applied associate-*l/_binary64_126380.4

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

  15. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{x}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}}}{x + y}\]
  16. Using strategy rm

  17. Applied associate-/r/_binary64_126410.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2021093 
(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))))