Average Error: 0.4 → 0.5
Time: 14.5s
Precision: 64
\[\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)\]
↓
\[\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{x}}{\sqrt[3]{9.0}} \cdot e^{\log \left(\frac{1.0}{\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}}\right)}\right) - 1.0\right)\]
\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)
↓
\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{x}}{\sqrt[3]{9.0}} \cdot e^{\log \left(\frac{1.0}{\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}}\right)}\right) - 1.0\right)
double f(double x, double y) {
        double r18967991 = 3.0;
        double r18967992 = x;
        double r18967993 = sqrt(r18967992);
        double r18967994 = r18967991 * r18967993;
        double r18967995 = y;
        double r18967996 = 1.0;
        double r18967997 = 9.0;
        double r18967998 = r18967992 * r18967997;
        double r18967999 = r18967996 / r18967998;
        double r18968000 = r18967995 + r18967999;
        double r18968001 = r18968000 - r18967996;
        double r18968002 = r18967994 * r18968001;
        return r18968002;
}
↓
double f(double x, double y) {
        double r18968003 = 3.0;
        double r18968004 = x;
        double r18968005 = sqrt(r18968004);
        double r18968006 = r18968003 * r18968005;
        double r18968007 = y;
        double r18968008 = 1.0;
        double r18968009 = r18968008 / r18968004;
        double r18968010 = 9.0;
        double r18968011 = cbrt(r18968010);
        double r18968012 = r18968009 / r18968011;
        double r18968013 = 1.0;
        double r18968014 = r18968011 * r18968011;
        double r18968015 = r18968013 / r18968014;
        double r18968016 = log(r18968015);
        double r18968017 = exp(r18968016);
        double r18968018 = r18968012 * r18968017;
        double r18968019 = r18968007 + r18968018;
        double r18968020 = r18968019 - r18968013;
        double r18968021 = r18968006 * r18968020;
        return r18968021;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.5
\[3.0 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1.0}{x \cdot 9.0} - 1.0\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)\]
  2. Using strategy rm

  3. Applied associate-/r*0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{\frac{1.0}{x}}{9.0}}\right) - 1.0\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1.0}{x}}{\color{blue}{\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}}}\right) - 1.0\right)\]

  6. Applied div-inv0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\color{blue}{1.0 \cdot \frac{1}{x}}}{\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}}\right) - 1.0\right)\]

  7. Applied times-frac0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{1.0}{\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}}\right) - 1.0\right)\]
  8. Using strategy rm

  9. Applied add-exp-log0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{\sqrt[3]{9.0} \cdot \color{blue}{e^{\log \left(\sqrt[3]{9.0}\right)}}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  10. Applied add-exp-log0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{\color{blue}{e^{\log \left(\sqrt[3]{9.0}\right)}} \cdot e^{\log \left(\sqrt[3]{9.0}\right)}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  11. Applied prod-exp0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{\color{blue}{e^{\log \left(\sqrt[3]{9.0}\right) + \log \left(\sqrt[3]{9.0}\right)}}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  12. Applied add-exp-log0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\color{blue}{e^{\log 1.0}}}{e^{\log \left(\sqrt[3]{9.0}\right) + \log \left(\sqrt[3]{9.0}\right)}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  13. Applied div-exp0.5

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{e^{\log 1.0 - \left(\log \left(\sqrt[3]{9.0}\right) + \log \left(\sqrt[3]{9.0}\right)\right)}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  14. Simplified0.5

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + e^{\color{blue}{\log \left(\frac{1.0}{\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}}\right)}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9.0}}\right) - 1.0\right)\]

  15. Final simplification0.5

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{x}}{\sqrt[3]{9.0}} \cdot e^{\log \left(\frac{1.0}{\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}}\right)}\right) - 1.0\right)\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

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

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