Average Error: 9.2 → 0.1
Time: 12.3s
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 r562475 = x;
        double r562476 = y;
        double r562477 = r562475 / r562476;
        double r562478 = 1.0;
        double r562479 = r562477 + r562478;
        double r562480 = r562475 * r562479;
        double r562481 = r562475 + r562478;
        double r562482 = r562480 / r562481;
        return r562482;
}
↓
double f(double x, double y) {
        double r562483 = x;
        double r562484 = 1.0;
        double r562485 = r562483 + r562484;
        double r562486 = y;
        double r562487 = r562483 / r562486;
        double r562488 = r562487 + r562484;
        double r562489 = r562485 / r562488;
        double r562490 = r562483 / r562489;
        return r562490;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 2019325 +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)))