Average Error: 9.3 → 0.1
Time: 12.9s
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 r499394 = x;
        double r499395 = y;
        double r499396 = r499394 / r499395;
        double r499397 = 1.0;
        double r499398 = r499396 + r499397;
        double r499399 = r499394 * r499398;
        double r499400 = r499394 + r499397;
        double r499401 = r499399 / r499400;
        return r499401;
}
↓
double f(double x, double y) {
        double r499402 = x;
        double r499403 = 1.0;
        double r499404 = r499402 + r499403;
        double r499405 = y;
        double r499406 = r499402 / r499405;
        double r499407 = r499406 + r499403;
        double r499408 = r499404 / r499407;
        double r499409 = r499402 / r499408;
        return r499409;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019212 +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)))