Average Error: 9.2 → 0.1
Time: 14.7s
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 r576976 = x;
        double r576977 = y;
        double r576978 = r576976 / r576977;
        double r576979 = 1.0;
        double r576980 = r576978 + r576979;
        double r576981 = r576976 * r576980;
        double r576982 = r576976 + r576979;
        double r576983 = r576981 / r576982;
        return r576983;
}
↓
double f(double x, double y) {
        double r576984 = x;
        double r576985 = 1.0;
        double r576986 = r576984 + r576985;
        double r576987 = y;
        double r576988 = r576984 / r576987;
        double r576989 = r576988 + r576985;
        double r576990 = r576986 / r576989;
        double r576991 = r576984 / r576990;
        return r576991;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019323 +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)))