Average Error: 9.0 → 0.1
Time: 3.0s
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 r948664 = x;
        double r948665 = y;
        double r948666 = r948664 / r948665;
        double r948667 = 1.0;
        double r948668 = r948666 + r948667;
        double r948669 = r948664 * r948668;
        double r948670 = r948664 + r948667;
        double r948671 = r948669 / r948670;
        return r948671;
}
↓
double f(double x, double y) {
        double r948672 = x;
        double r948673 = 1.0;
        double r948674 = r948672 + r948673;
        double r948675 = y;
        double r948676 = r948672 / r948675;
        double r948677 = r948676 + r948673;
        double r948678 = r948674 / r948677;
        double r948679 = r948672 / r948678;
        return r948679;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020060 
(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)))