Average Error: 9.1 → 0.1
Time: 10.7s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[x \cdot \frac{\frac{x}{y} + 1}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
x \cdot \frac{\frac{x}{y} + 1}{x + 1}
double f(double x, double y) {
        double r977851 = x;
        double r977852 = y;
        double r977853 = r977851 / r977852;
        double r977854 = 1.0;
        double r977855 = r977853 + r977854;
        double r977856 = r977851 * r977855;
        double r977857 = r977851 + r977854;
        double r977858 = r977856 / r977857;
        return r977858;
}
↓
double f(double x, double y) {
        double r977859 = x;
        double r977860 = y;
        double r977861 = r977859 / r977860;
        double r977862 = 1.0;
        double r977863 = r977861 + r977862;
        double r977864 = r977859 + r977862;
        double r977865 = r977863 / r977864;
        double r977866 = r977859 * r977865;
        return r977866;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity9.1

    \[\leadsto \frac{x \cdot \left(\frac{x}{y} + 1\right)}{\color{blue}{1 \cdot \left(x + 1\right)}}\]

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{x} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

  6. Final simplification0.1

    \[\leadsto x \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Reproduce

herbie shell --seed 2020047 
(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)))