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 r552447 = x;
        double r552448 = y;
        double r552449 = r552447 / r552448;
        double r552450 = 1.0;
        double r552451 = r552449 + r552450;
        double r552452 = r552447 * r552451;
        double r552453 = r552447 + r552450;
        double r552454 = r552452 / r552453;
        return r552454;
}
↓
double f(double x, double y) {
        double r552455 = x;
        double r552456 = 1.0;
        double r552457 = r552455 + r552456;
        double r552458 = y;
        double r552459 = r552455 / r552458;
        double r552460 = r552459 + r552456;
        double r552461 = r552457 / r552460;
        double r552462 = r552455 / r552461;
        return r552462;
}

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 
(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)))