Average Error: 0.2 → 0.2
Time: 9.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{9 \cdot x}\right) - \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\frac{\sqrt{x}}{\frac{y}{\sqrt[3]{3}}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{9 \cdot x}\right) - \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\frac{\sqrt{x}}{\frac{y}{\sqrt[3]{3}}}}
double f(double x, double y) {
        double r548224 = 1.0;
        double r548225 = x;
        double r548226 = 9.0;
        double r548227 = r548225 * r548226;
        double r548228 = r548224 / r548227;
        double r548229 = r548224 - r548228;
        double r548230 = y;
        double r548231 = 3.0;
        double r548232 = sqrt(r548225);
        double r548233 = r548231 * r548232;
        double r548234 = r548230 / r548233;
        double r548235 = r548229 - r548234;
        return r548235;
}
↓
double f(double x, double y) {
        double r548236 = 1.0;
        double r548237 = 9.0;
        double r548238 = x;
        double r548239 = r548237 * r548238;
        double r548240 = r548236 / r548239;
        double r548241 = r548236 - r548240;
        double r548242 = 1.0;
        double r548243 = 3.0;
        double r548244 = cbrt(r548243);
        double r548245 = r548244 * r548244;
        double r548246 = r548242 / r548245;
        double r548247 = sqrt(r548238);
        double r548248 = y;
        double r548249 = r548248 / r548244;
        double r548250 = r548247 / r548249;
        double r548251 = r548246 / r548250;
        double r548252 = r548241 - r548251;
        return r548252;
}

Error

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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. 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 associate-/r*0.2

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

  7. Applied add-cube-cbrt0.2

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

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

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

  9. Applied times-frac0.3

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

  10. Applied associate-/l*0.3

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

  12. Applied associate-/l/0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020057 
(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)))))