Average Error: 8.5 → 0.1
Time: 11.7s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}
double f(double x, double y) {
        double r28386456 = x;
        double r28386457 = y;
        double r28386458 = r28386456 / r28386457;
        double r28386459 = 1.0;
        double r28386460 = r28386458 + r28386459;
        double r28386461 = r28386456 * r28386460;
        double r28386462 = r28386456 + r28386459;
        double r28386463 = r28386461 / r28386462;
        return r28386463;
}
↓
double f(double x, double y) {
        double r28386464 = x;
        double r28386465 = 1.0;
        double r28386466 = r28386465 + r28386464;
        double r28386467 = y;
        double r28386468 = r28386464 / r28386467;
        double r28386469 = r28386465 + r28386468;
        double r28386470 = r28386466 / r28386469;
        double r28386471 = r28386464 / r28386470;
        return r28386471;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  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. Final simplification0.1

    \[\leadsto \frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019163 +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)))