Average Error: 9.3 → 0.1
Time: 16.5s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)
double f(double x, double y) {
        double r596193 = x;
        double r596194 = y;
        double r596195 = r596193 / r596194;
        double r596196 = 1.0;
        double r596197 = r596195 + r596196;
        double r596198 = r596193 * r596197;
        double r596199 = r596193 + r596196;
        double r596200 = r596198 / r596199;
        return r596200;
}
↓
double f(double x, double y) {
        double r596201 = x;
        double r596202 = 1.0;
        double r596203 = r596201 + r596202;
        double r596204 = r596201 / r596203;
        double r596205 = y;
        double r596206 = r596201 / r596205;
        double r596207 = r596206 + r596202;
        double r596208 = r596204 * r596207;
        return r596208;
}

Error

Bits error versus x

Bits error versus y

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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. Using strategy rm

  5. Applied associate-/r/0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019199 +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)))