Average Error: 9.6 → 0.1
Time: 5.7s
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 r495380 = x;
        double r495381 = y;
        double r495382 = r495380 / r495381;
        double r495383 = 1.0;
        double r495384 = r495382 + r495383;
        double r495385 = r495380 * r495384;
        double r495386 = r495380 + r495383;
        double r495387 = r495385 / r495386;
        return r495387;
}
↓
double f(double x, double y) {
        double r495388 = x;
        double r495389 = 1.0;
        double r495390 = r495388 + r495389;
        double r495391 = y;
        double r495392 = r495388 / r495391;
        double r495393 = r495392 + r495389;
        double r495394 = r495390 / r495393;
        double r495395 = r495388 / r495394;
        return r495395;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019304 
(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)))