Average Error: 8.7 → 0.1
Time: 7.4s
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 r16464586 = x;
        double r16464587 = y;
        double r16464588 = r16464586 / r16464587;
        double r16464589 = 1.0;
        double r16464590 = r16464588 + r16464589;
        double r16464591 = r16464586 * r16464590;
        double r16464592 = r16464586 + r16464589;
        double r16464593 = r16464591 / r16464592;
        return r16464593;
}
↓
double f(double x, double y) {
        double r16464594 = x;
        double r16464595 = 1.0;
        double r16464596 = r16464595 + r16464594;
        double r16464597 = r16464594 / r16464596;
        double r16464598 = y;
        double r16464599 = r16464594 / r16464598;
        double r16464600 = r16464595 + r16464599;
        double r16464601 = r16464597 * r16464600;
        return r16464601;
}

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

Original8.7
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.7

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  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 2019156 +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)))