Average Error: 0.2 → 0.3
Time: 5.4s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\sqrt{1}}{1} \cdot \frac{\frac{y}{\sqrt{x}}}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\sqrt{1}}{1} \cdot \frac{\frac{y}{\sqrt{x}}}{3}
double f(double x, double y) {
        double r333256 = 1.0;
        double r333257 = x;
        double r333258 = 9.0;
        double r333259 = r333257 * r333258;
        double r333260 = r333256 / r333259;
        double r333261 = r333256 - r333260;
        double r333262 = y;
        double r333263 = 3.0;
        double r333264 = sqrt(r333257);
        double r333265 = r333263 * r333264;
        double r333266 = r333262 / r333265;
        double r333267 = r333261 - r333266;
        return r333267;
}
↓
double f(double x, double y) {
        double r333268 = 1.0;
        double r333269 = x;
        double r333270 = r333268 / r333269;
        double r333271 = 9.0;
        double r333272 = r333270 / r333271;
        double r333273 = r333268 - r333272;
        double r333274 = 1.0;
        double r333275 = sqrt(r333274);
        double r333276 = r333275 / r333274;
        double r333277 = y;
        double r333278 = sqrt(r333269);
        double r333279 = r333277 / r333278;
        double r333280 = 3.0;
        double r333281 = r333279 / r333280;
        double r333282 = r333276 * r333281;
        double r333283 = r333273 - r333282;
        return r333283;
}

Error

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Results

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Target

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

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.3

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{9}\right) - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

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

  9. Applied add-sqr-sqrt0.3

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

  10. Applied times-frac0.3

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

  11. Applied associate-*l*0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(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)))))