Average Error: 0.2 → 0.2
Time: 18.2s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 + \frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{9}{\sqrt[3]{-1}} \cdot x}\right) - \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt{x} \cdot \sqrt[3]{3}} \cdot y\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 + \frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{9}{\sqrt[3]{-1}} \cdot x}\right) - \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt{x} \cdot \sqrt[3]{3}} \cdot y
double f(double x, double y) {
        double r528279 = 1.0;
        double r528280 = x;
        double r528281 = 9.0;
        double r528282 = r528280 * r528281;
        double r528283 = r528279 / r528282;
        double r528284 = r528279 - r528283;
        double r528285 = y;
        double r528286 = 3.0;
        double r528287 = sqrt(r528280);
        double r528288 = r528286 * r528287;
        double r528289 = r528285 / r528288;
        double r528290 = r528284 - r528289;
        return r528290;
}
↓
double f(double x, double y) {
        double r528291 = 1.0;
        double r528292 = -r528291;
        double r528293 = cbrt(r528292);
        double r528294 = r528293 * r528293;
        double r528295 = 9.0;
        double r528296 = r528295 / r528293;
        double r528297 = x;
        double r528298 = r528296 * r528297;
        double r528299 = r528294 / r528298;
        double r528300 = r528291 + r528299;
        double r528301 = 1.0;
        double r528302 = 3.0;
        double r528303 = cbrt(r528302);
        double r528304 = r528303 * r528303;
        double r528305 = r528301 / r528304;
        double r528306 = sqrt(r528297);
        double r528307 = r528306 * r528303;
        double r528308 = r528305 / r528307;
        double r528309 = y;
        double r528310 = r528308 * r528309;
        double r528311 = r528300 - r528310;
        return r528311;
}

Error

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Bits error versus y

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Results

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Target

Original0.2
Target0.3
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. Simplified0.2

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

  4. Applied sub-neg0.2

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

  5. Simplified0.3

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

  7. Applied div-inv0.3

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

  8. Simplified0.3

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

  10. Applied add-cube-cbrt0.3

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

  11. Applied add-sqr-sqrt0.3

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

  12. Applied times-frac0.4

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

  13. Applied associate-/l*0.3

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

  14. Simplified0.3

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

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

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

  17. Applied add-cube-cbrt0.3

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

  18. Applied times-frac0.3

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

  19. Applied associate-/l*0.3

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

  20. Simplified0.2

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

  21. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))