Average Error: 9.2 → 0.1
Time: 3.1s
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 r939818 = x;
        double r939819 = y;
        double r939820 = r939818 / r939819;
        double r939821 = 1.0;
        double r939822 = r939820 + r939821;
        double r939823 = r939818 * r939822;
        double r939824 = r939818 + r939821;
        double r939825 = r939823 / r939824;
        return r939825;
}
↓
double f(double x, double y) {
        double r939826 = x;
        double r939827 = 1.0;
        double r939828 = r939826 + r939827;
        double r939829 = y;
        double r939830 = r939826 / r939829;
        double r939831 = r939830 + r939827;
        double r939832 = r939828 / r939831;
        double r939833 = r939826 / r939832;
        return r939833;
}

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 2020100 
(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)))