Average Error: 9.0 → 0.1
Time: 3.0s
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 r1200903 = x;
        double r1200904 = y;
        double r1200905 = r1200903 / r1200904;
        double r1200906 = 1.0;
        double r1200907 = r1200905 + r1200906;
        double r1200908 = r1200903 * r1200907;
        double r1200909 = r1200903 + r1200906;
        double r1200910 = r1200908 / r1200909;
        return r1200910;
}
↓
double f(double x, double y) {
        double r1200911 = x;
        double r1200912 = 1.0;
        double r1200913 = r1200911 + r1200912;
        double r1200914 = y;
        double r1200915 = r1200911 / r1200914;
        double r1200916 = r1200915 + r1200912;
        double r1200917 = r1200913 / r1200916;
        double r1200918 = r1200911 / r1200917;
        return r1200918;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020034 +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)))