Average Error: 9.4 → 0.1
Time: 11.1s
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 r605986 = x;
        double r605987 = y;
        double r605988 = r605986 / r605987;
        double r605989 = 1.0;
        double r605990 = r605988 + r605989;
        double r605991 = r605986 * r605990;
        double r605992 = r605986 + r605989;
        double r605993 = r605991 / r605992;
        return r605993;
}
↓
double f(double x, double y) {
        double r605994 = x;
        double r605995 = 1.0;
        double r605996 = r605994 + r605995;
        double r605997 = y;
        double r605998 = r605994 / r605997;
        double r605999 = r605998 + r605995;
        double r606000 = r605996 / r605999;
        double r606001 = r605994 / r606000;
        return r606001;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020043 +o rules:numerics
(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)))