Average Error: 9.3 → 0.1
Time: 5.1s
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 r895334 = x;
        double r895335 = y;
        double r895336 = r895334 / r895335;
        double r895337 = 1.0;
        double r895338 = r895336 + r895337;
        double r895339 = r895334 * r895338;
        double r895340 = r895334 + r895337;
        double r895341 = r895339 / r895340;
        return r895341;
}
↓
double f(double x, double y) {
        double r895342 = x;
        double r895343 = 1.0;
        double r895344 = r895342 + r895343;
        double r895345 = y;
        double r895346 = r895342 / r895345;
        double r895347 = r895346 + r895343;
        double r895348 = r895344 / r895347;
        double r895349 = r895342 / r895348;
        return r895349;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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 2020021 +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)))