Average Error: 9.4 → 0.1
Time: 10.3s
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 r776790 = x;
        double r776791 = y;
        double r776792 = r776790 / r776791;
        double r776793 = 1.0;
        double r776794 = r776792 + r776793;
        double r776795 = r776790 * r776794;
        double r776796 = r776790 + r776793;
        double r776797 = r776795 / r776796;
        return r776797;
}
↓
double f(double x, double y) {
        double r776798 = x;
        double r776799 = 1.0;
        double r776800 = r776798 + r776799;
        double r776801 = y;
        double r776802 = r776798 / r776801;
        double r776803 = r776802 + r776799;
        double r776804 = r776800 / r776803;
        double r776805 = r776798 / r776804;
        return r776805;
}

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 
(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)))