Average Error: 0.2 → 0.2
Time: 12.8s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r227229 = 1.0;
        double r227230 = x;
        double r227231 = 9.0;
        double r227232 = r227230 * r227231;
        double r227233 = r227229 / r227232;
        double r227234 = r227229 - r227233;
        double r227235 = y;
        double r227236 = 3.0;
        double r227237 = sqrt(r227230);
        double r227238 = r227236 * r227237;
        double r227239 = r227235 / r227238;
        double r227240 = r227234 - r227239;
        return r227240;
}
↓
double f(double x, double y) {
        double r227241 = 1.0;
        double r227242 = 0.1111111111111111;
        double r227243 = x;
        double r227244 = r227242 / r227243;
        double r227245 = r227241 - r227244;
        double r227246 = y;
        double r227247 = 3.0;
        double r227248 = sqrt(r227243);
        double r227249 = r227247 * r227248;
        double r227250 = r227246 / r227249;
        double r227251 = r227245 - r227250;
        return r227251;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  3. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))