Average Error: 9.3 → 0.1
Time: 12.8s
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 r555658 = x;
        double r555659 = y;
        double r555660 = r555658 / r555659;
        double r555661 = 1.0;
        double r555662 = r555660 + r555661;
        double r555663 = r555658 * r555662;
        double r555664 = r555658 + r555661;
        double r555665 = r555663 / r555664;
        return r555665;
}
↓
double f(double x, double y) {
        double r555666 = x;
        double r555667 = 1.0;
        double r555668 = r555666 + r555667;
        double r555669 = y;
        double r555670 = r555666 / r555669;
        double r555671 = r555670 + r555667;
        double r555672 = r555668 / r555671;
        double r555673 = r555666 / r555672;
        return r555673;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019235 +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)))