Average Error: 9.3 → 0.1
Time: 13.0s
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 r513341 = x;
        double r513342 = y;
        double r513343 = r513341 / r513342;
        double r513344 = 1.0;
        double r513345 = r513343 + r513344;
        double r513346 = r513341 * r513345;
        double r513347 = r513341 + r513344;
        double r513348 = r513346 / r513347;
        return r513348;
}
↓
double f(double x, double y) {
        double r513349 = x;
        double r513350 = 1.0;
        double r513351 = r513349 + r513350;
        double r513352 = y;
        double r513353 = r513349 / r513352;
        double r513354 = r513353 + r513350;
        double r513355 = r513351 / r513354;
        double r513356 = r513349 / r513355;
        return r513356;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019303 +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)))