Average Error: 9.3 → 0.1
Time: 10.9s
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 r788429 = x;
        double r788430 = y;
        double r788431 = r788429 / r788430;
        double r788432 = 1.0;
        double r788433 = r788431 + r788432;
        double r788434 = r788429 * r788433;
        double r788435 = r788429 + r788432;
        double r788436 = r788434 / r788435;
        return r788436;
}
↓
double f(double x, double y) {
        double r788437 = x;
        double r788438 = 1.0;
        double r788439 = r788437 + r788438;
        double r788440 = y;
        double r788441 = r788437 / r788440;
        double r788442 = r788441 + r788438;
        double r788443 = r788439 / r788442;
        double r788444 = r788437 / r788443;
        return r788444;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(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)))