Average Error: 8.6 → 0.1
Time: 10.6s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)
double f(double x, double y) {
        double r43482934 = x;
        double r43482935 = y;
        double r43482936 = r43482934 / r43482935;
        double r43482937 = 1.0;
        double r43482938 = r43482936 + r43482937;
        double r43482939 = r43482934 * r43482938;
        double r43482940 = r43482934 + r43482937;
        double r43482941 = r43482939 / r43482940;
        return r43482941;
}
↓
double f(double x, double y) {
        double r43482942 = x;
        double r43482943 = 1.0;
        double r43482944 = r43482943 + r43482942;
        double r43482945 = r43482942 / r43482944;
        double r43482946 = y;
        double r43482947 = r43482942 / r43482946;
        double r43482948 = r43482943 + r43482947;
        double r43482949 = r43482945 * r43482948;
        return r43482949;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original8.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1.0} \cdot \left(\frac{x}{y} + 1.0\right)}\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))