Average Error: 9.4 → 0.1
Time: 14.0s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}
double f(double x, double y) {
        double r946351 = x;
        double r946352 = y;
        double r946353 = r946351 / r946352;
        double r946354 = 1.0;
        double r946355 = r946353 + r946354;
        double r946356 = r946351 * r946355;
        double r946357 = r946351 + r946354;
        double r946358 = r946356 / r946357;
        return r946358;
}
↓
double f(double x, double y) {
        double r946359 = x;
        double r946360 = 1.0;
        double r946361 = r946359 + r946360;
        double r946362 = y;
        double r946363 = r946359 / r946362;
        double r946364 = r946363 + r946360;
        double r946365 = r946361 / r946364;
        double r946366 = r946359 / r946365;
        return r946366;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +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)))