Average Error: 9.2 → 0.1
Time: 14.5s
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 r629454 = x;
        double r629455 = y;
        double r629456 = r629454 / r629455;
        double r629457 = 1.0;
        double r629458 = r629456 + r629457;
        double r629459 = r629454 * r629458;
        double r629460 = r629454 + r629457;
        double r629461 = r629459 / r629460;
        return r629461;
}
↓
double f(double x, double y) {
        double r629462 = x;
        double r629463 = 1.0;
        double r629464 = r629462 + r629463;
        double r629465 = y;
        double r629466 = r629462 / r629465;
        double r629467 = r629466 + r629463;
        double r629468 = r629464 / r629467;
        double r629469 = r629462 / r629468;
        return r629469;
}

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 +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)))