Average Error: 9.2 → 0.1
Time: 8.2s
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 r948853 = x;
        double r948854 = y;
        double r948855 = r948853 / r948854;
        double r948856 = 1.0;
        double r948857 = r948855 + r948856;
        double r948858 = r948853 * r948857;
        double r948859 = r948853 + r948856;
        double r948860 = r948858 / r948859;
        return r948860;
}
↓
double f(double x, double y) {
        double r948861 = x;
        double r948862 = 1.0;
        double r948863 = r948861 + r948862;
        double r948864 = y;
        double r948865 = r948861 / r948864;
        double r948866 = r948865 + r948862;
        double r948867 = r948863 / r948866;
        double r948868 = r948861 / r948867;
        return r948868;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020042 
(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)))