Average Error: 8.6 → 0.1
Time: 12.1s
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 r28672395 = x;
        double r28672396 = y;
        double r28672397 = r28672395 / r28672396;
        double r28672398 = 1.0;
        double r28672399 = r28672397 + r28672398;
        double r28672400 = r28672395 * r28672399;
        double r28672401 = r28672395 + r28672398;
        double r28672402 = r28672400 / r28672401;
        return r28672402;
}
↓
double f(double x, double y) {
        double r28672403 = x;
        double r28672404 = 1.0;
        double r28672405 = r28672404 + r28672403;
        double r28672406 = r28672403 / r28672405;
        double r28672407 = y;
        double r28672408 = r28672403 / r28672407;
        double r28672409 = r28672404 + r28672408;
        double r28672410 = r28672406 * r28672409;
        return r28672410;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original8.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  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 2019168 +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)))