Average Error: 9.3 → 0.1
Time: 13.1s
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 r589027 = x;
        double r589028 = y;
        double r589029 = r589027 / r589028;
        double r589030 = 1.0;
        double r589031 = r589029 + r589030;
        double r589032 = r589027 * r589031;
        double r589033 = r589027 + r589030;
        double r589034 = r589032 / r589033;
        return r589034;
}
↓
double f(double x, double y) {
        double r589035 = x;
        double r589036 = 1.0;
        double r589037 = r589035 + r589036;
        double r589038 = y;
        double r589039 = r589035 / r589038;
        double r589040 = r589039 + r589036;
        double r589041 = r589037 / r589040;
        double r589042 = r589035 / r589041;
        return r589042;
}

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 2019303 
(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)))