Average Error: 9.6 → 0.1
Time: 13.3s
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 r565942 = x;
        double r565943 = y;
        double r565944 = r565942 / r565943;
        double r565945 = 1.0;
        double r565946 = r565944 + r565945;
        double r565947 = r565942 * r565946;
        double r565948 = r565942 + r565945;
        double r565949 = r565947 / r565948;
        return r565949;
}
↓
double f(double x, double y) {
        double r565950 = x;
        double r565951 = 1.0;
        double r565952 = r565950 + r565951;
        double r565953 = y;
        double r565954 = r565950 / r565953;
        double r565955 = r565954 + r565951;
        double r565956 = r565952 / r565955;
        double r565957 = r565950 / r565956;
        return r565957;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019304 
(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)))