Average Error: 9.2 → 0.1
Time: 12.0s
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 r600175 = x;
        double r600176 = y;
        double r600177 = r600175 / r600176;
        double r600178 = 1.0;
        double r600179 = r600177 + r600178;
        double r600180 = r600175 * r600179;
        double r600181 = r600175 + r600178;
        double r600182 = r600180 / r600181;
        return r600182;
}
↓
double f(double x, double y) {
        double r600183 = x;
        double r600184 = 1.0;
        double r600185 = r600183 + r600184;
        double r600186 = y;
        double r600187 = r600183 / r600186;
        double r600188 = r600187 + r600184;
        double r600189 = r600185 / r600188;
        double r600190 = r600183 / r600189;
        return r600190;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019325 
(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)))