Average Error: 8.4 → 0.1
Time: 9.5s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)
double f(double x, double y) {
        double r36399336 = x;
        double r36399337 = y;
        double r36399338 = r36399336 / r36399337;
        double r36399339 = 1.0;
        double r36399340 = r36399338 + r36399339;
        double r36399341 = r36399336 * r36399340;
        double r36399342 = r36399336 + r36399339;
        double r36399343 = r36399341 / r36399342;
        return r36399343;
}
↓
double f(double x, double y) {
        double r36399344 = x;
        double r36399345 = 1.0;
        double r36399346 = r36399345 + r36399344;
        double r36399347 = r36399344 / r36399346;
        double r36399348 = y;
        double r36399349 = r36399344 / r36399348;
        double r36399350 = r36399345 + r36399349;
        double r36399351 = r36399347 * r36399350;
        return r36399351;
}

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

Original8.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1.0} \cdot \left(\frac{x}{y} + 1.0\right)}\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))