Average Error: 8.9 → 0.1
Time: 14.9s
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 r525736 = x;
        double r525737 = y;
        double r525738 = r525736 / r525737;
        double r525739 = 1.0;
        double r525740 = r525738 + r525739;
        double r525741 = r525736 * r525740;
        double r525742 = r525736 + r525739;
        double r525743 = r525741 / r525742;
        return r525743;
}
↓
double f(double x, double y) {
        double r525744 = x;
        double r525745 = 1.0;
        double r525746 = r525744 + r525745;
        double r525747 = y;
        double r525748 = r525744 / r525747;
        double r525749 = r525748 + r525745;
        double r525750 = r525746 / r525749;
        double r525751 = r525744 / r525750;
        return r525751;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original8.9
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.9

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019326 +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)))