Average Error: 9.4 → 0.1
Time: 4.7s
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 r874369 = x;
        double r874370 = y;
        double r874371 = r874369 / r874370;
        double r874372 = 1.0;
        double r874373 = r874371 + r874372;
        double r874374 = r874369 * r874373;
        double r874375 = r874369 + r874372;
        double r874376 = r874374 / r874375;
        return r874376;
}
↓
double f(double x, double y) {
        double r874377 = x;
        double r874378 = 1.0;
        double r874379 = r874377 + r874378;
        double r874380 = y;
        double r874381 = r874377 / r874380;
        double r874382 = r874381 + r874378;
        double r874383 = r874379 / r874382;
        double r874384 = r874377 / r874383;
        return r874384;
}

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 2020046 
(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)))