Average Error: 9.0 → 0.1
Time: 13.3s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}
double f(double x, double y) {
        double r727483 = x;
        double r727484 = y;
        double r727485 = r727483 / r727484;
        double r727486 = 1.0;
        double r727487 = r727485 + r727486;
        double r727488 = r727483 * r727487;
        double r727489 = r727483 + r727486;
        double r727490 = r727488 / r727489;
        return r727490;
}
↓
double f(double x, double y) {
        double r727491 = x;
        double r727492 = y;
        double r727493 = r727491 / r727492;
        double r727494 = 1.0;
        double r727495 = r727493 + r727494;
        double r727496 = r727491 + r727494;
        double r727497 = r727491 / r727496;
        double r727498 = r727495 * r727497;
        return r727498;
}

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

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1 + \frac{x}{y}}{\frac{1 + x}{x}}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) \cdot \frac{1}{\frac{1 + x}{x}}}\]

  5. Simplified0.1

    \[\leadsto \left(1 + \frac{x}{y}\right) \cdot \color{blue}{\frac{1 \cdot x}{1 + x}}\]

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

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

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