Average Error: 9.0 → 0.2
Time: 6.6s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{\frac{x}{y} + 1}{\mathsf{fma}\left(1, \frac{1}{x}, 1\right)}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{\frac{x}{y} + 1}{\mathsf{fma}\left(1, \frac{1}{x}, 1\right)}
double code(double x, double y) {
	return ((x * ((x / y) + 1.0)) / (x + 1.0));
}
↓
double code(double x, double y) {
	return (((x / y) + 1.0) / fma(1.0, (1.0 / x), 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

Original9.0
Target0.1
Herbie0.2
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

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

  3. Applied *-commutative9.0

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

  4. Applied associate-/l*0.2

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

    \[\leadsto \frac{\frac{x}{y} + 1}{\mathsf{fma}\left(1, \frac{1}{x}, 1\right)}\]

Reproduce

herbie shell --seed 2020066 +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)))