Average Error: 9.0 → 0.1
Time: 3.6s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (x / y)) + 1.0)) * ((double) (x / ((double) (x + 1.0))))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

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

  3. Applied *-un-lft-identity9.0

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

  4. Applied *-commutative9.0

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020114 
(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)))