Average Error: 9.5 → 0.1
Time: 3.6s
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 r1061786 = x;
        double r1061787 = y;
        double r1061788 = r1061786 / r1061787;
        double r1061789 = 1.0;
        double r1061790 = r1061788 + r1061789;
        double r1061791 = r1061786 * r1061790;
        double r1061792 = r1061786 + r1061789;
        double r1061793 = r1061791 / r1061792;
        return r1061793;
}
↓
double f(double x, double y) {
        double r1061794 = x;
        double r1061795 = 1.0;
        double r1061796 = r1061794 + r1061795;
        double r1061797 = y;
        double r1061798 = r1061794 / r1061797;
        double r1061799 = r1061798 + r1061795;
        double r1061800 = r1061796 / r1061799;
        double r1061801 = r1061794 / r1061800;
        return r1061801;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019353 +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)))