Average Error: 9.3 → 0.1
Time: 13.4s
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 r551771 = x;
        double r551772 = y;
        double r551773 = r551771 / r551772;
        double r551774 = 1.0;
        double r551775 = r551773 + r551774;
        double r551776 = r551771 * r551775;
        double r551777 = r551771 + r551774;
        double r551778 = r551776 / r551777;
        return r551778;
}
↓
double f(double x, double y) {
        double r551779 = x;
        double r551780 = 1.0;
        double r551781 = r551779 + r551780;
        double r551782 = y;
        double r551783 = r551779 / r551782;
        double r551784 = r551783 + r551780;
        double r551785 = r551781 / r551784;
        double r551786 = r551779 / r551785;
        return r551786;
}

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 +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)))