Average Error: 8.9 → 0.1
Time: 5.9s
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 r855680 = x;
        double r855681 = y;
        double r855682 = r855680 / r855681;
        double r855683 = 1.0;
        double r855684 = r855682 + r855683;
        double r855685 = r855680 * r855684;
        double r855686 = r855680 + r855683;
        double r855687 = r855685 / r855686;
        return r855687;
}
↓
double f(double x, double y) {
        double r855688 = x;
        double r855689 = 1.0;
        double r855690 = r855688 + r855689;
        double r855691 = y;
        double r855692 = r855688 / r855691;
        double r855693 = r855692 + r855689;
        double r855694 = r855690 / r855693;
        double r855695 = r855688 / r855694;
        return r855695;
}

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

Original8.9
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.9

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020089 +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)))