Average Error: 9.5 → 0.1
Time: 2.6s
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 r953253 = x;
        double r953254 = y;
        double r953255 = r953253 / r953254;
        double r953256 = 1.0;
        double r953257 = r953255 + r953256;
        double r953258 = r953253 * r953257;
        double r953259 = r953253 + r953256;
        double r953260 = r953258 / r953259;
        return r953260;
}
↓
double f(double x, double y) {
        double r953261 = x;
        double r953262 = 1.0;
        double r953263 = r953261 + r953262;
        double r953264 = y;
        double r953265 = r953261 / r953264;
        double r953266 = r953265 + r953262;
        double r953267 = r953263 / r953266;
        double r953268 = r953261 / r953267;
        return r953268;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019353 
(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)))