Average Error: 8.6 → 0.1
Time: 1.8s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{-\left(x + 1\right)} \cdot \left(-\left(\frac{x}{y} + 1\right)\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{-\left(x + 1\right)} \cdot \left(-\left(\frac{x}{y} + 1\right)\right)
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)) * -((x / y) + 1.0));
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original8.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.6

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

  3. Applied associate-/l*0.1

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

  5. Applied frac-2neg0.1

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

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1)))

  (/ (* x (+ (/ x y) 1)) (+ x 1)))