Average Error: 9.4 → 0.1
Time: 4.4s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)
double f(double x, double y) {
        double r901363 = x;
        double r901364 = y;
        double r901365 = r901363 / r901364;
        double r901366 = 1.0;
        double r901367 = r901365 + r901366;
        double r901368 = r901363 * r901367;
        double r901369 = r901363 + r901366;
        double r901370 = r901368 / r901369;
        return r901370;
}
↓
double f(double x, double y) {
        double r901371 = x;
        double r901372 = 1.0;
        double r901373 = r901371 + r901372;
        double r901374 = r901371 / r901373;
        double r901375 = y;
        double r901376 = r901371 / r901375;
        double r901377 = r901376 + r901372;
        double r901378 = r901374 * r901377;
        return r901378;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 associate-/r/0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 +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)))