Average Error: 9.1 → 0.1
Time: 8.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \frac{x}{y} + 1 \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \frac{x}{y} + 1 \cdot \frac{x}{x + 1}
double f(double x, double y) {
        double r799499 = x;
        double r799500 = y;
        double r799501 = r799499 / r799500;
        double r799502 = 1.0;
        double r799503 = r799501 + r799502;
        double r799504 = r799499 * r799503;
        double r799505 = r799499 + r799502;
        double r799506 = r799504 / r799505;
        return r799506;
}
↓
double f(double x, double y) {
        double r799507 = x;
        double r799508 = 1.0;
        double r799509 = r799507 + r799508;
        double r799510 = r799507 / r799509;
        double r799511 = y;
        double r799512 = r799507 / r799511;
        double r799513 = r799510 * r799512;
        double r799514 = r799508 * r799510;
        double r799515 = r799513 + r799514;
        return r799515;
}

Error

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Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)}\]
  6. Using strategy rm

  7. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \frac{x}{y} + \frac{x}{x + 1} \cdot 1}\]

  8. Simplified0.1

    \[\leadsto \frac{x}{x + 1} \cdot \frac{x}{y} + \color{blue}{1 \cdot \frac{x}{x + 1}}\]

  9. Final simplification0.1

    \[\leadsto \frac{x}{x + 1} \cdot \frac{x}{y} + 1 \cdot \frac{x}{x + 1}\]

Reproduce

herbie shell --seed 2020047 +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)))