Average Error: 9.3 → 0.1
Time: 3.4s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)
double f(double x, double y) {
        double r723412 = x;
        double r723413 = y;
        double r723414 = r723412 / r723413;
        double r723415 = 1.0;
        double r723416 = r723414 + r723415;
        double r723417 = r723412 * r723416;
        double r723418 = r723412 + r723415;
        double r723419 = r723417 / r723418;
        return r723419;
}
↓
double f(double x, double y) {
        double r723420 = x;
        double r723421 = 1.0;
        double r723422 = r723420 + r723421;
        double r723423 = r723420 / r723422;
        double r723424 = y;
        double r723425 = r723420 / r723424;
        double r723426 = r723425 + r723421;
        double r723427 = r723423 * r723426;
        return r723427;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 
(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)))