Average Error: 9.4 → 0.1
Time: 9.3s
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 r637398 = x;
        double r637399 = y;
        double r637400 = r637398 / r637399;
        double r637401 = 1.0;
        double r637402 = r637400 + r637401;
        double r637403 = r637398 * r637402;
        double r637404 = r637398 + r637401;
        double r637405 = r637403 / r637404;
        return r637405;
}
↓
double f(double x, double y) {
        double r637406 = x;
        double r637407 = 1.0;
        double r637408 = r637406 + r637407;
        double r637409 = y;
        double r637410 = r637406 / r637409;
        double r637411 = r637410 + r637407;
        double r637412 = r637408 / r637411;
        double r637413 = r637406 / r637412;
        return r637413;
}

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 2019305 
(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)))