Average Error: 9.4 → 0.1
Time: 12.0s
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 r823289 = x;
        double r823290 = y;
        double r823291 = r823289 / r823290;
        double r823292 = 1.0;
        double r823293 = r823291 + r823292;
        double r823294 = r823289 * r823293;
        double r823295 = r823289 + r823292;
        double r823296 = r823294 / r823295;
        return r823296;
}
↓
double f(double x, double y) {
        double r823297 = x;
        double r823298 = 1.0;
        double r823299 = r823297 + r823298;
        double r823300 = r823297 / r823299;
        double r823301 = y;
        double r823302 = r823297 / r823301;
        double r823303 = r823302 + r823298;
        double r823304 = r823300 * r823303;
        return r823304;
}

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 2020046 +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)))