Average Error: 9.1 → 0.1
Time: 4.0s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right)
(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
↓
(FPCore (x y) :precision binary64 (* (/ x (+ x 1.0)) (+ 1.0 (/ x y))))
double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}
↓
double code(double x, double y) {
	return (x / (x + 1.0)) * (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

Original9.1
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.1

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

  3. Applied flip-+_binary6413.8

    \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{x}{y} \cdot \frac{x}{y} - 1 \cdot 1}{\frac{x}{y} - 1}}}{x + 1}\]

  4. Simplified13.8

    \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y} - 1}}{\frac{x}{y} - 1}}{x + 1}\]
  5. Using strategy rm

  6. Applied associate-/l*_binary6411.5

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{\frac{x}{y} \cdot \frac{x}{y} - 1}{\frac{x}{y} - 1}}}}\]

  7. Simplified0.1

    \[\leadsto \frac{x}{\color{blue}{\frac{1 + x}{1 + \frac{x}{y}}}}\]
  8. Using strategy rm

  9. Applied associate-/r/_binary640.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2021128 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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