Average Error: 0.4 → 0.5
Time: 7.9s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right)\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right)\right)\right)
double f(double x, double y) {
        double r421760 = 3.0;
        double r421761 = x;
        double r421762 = sqrt(r421761);
        double r421763 = r421760 * r421762;
        double r421764 = y;
        double r421765 = 1.0;
        double r421766 = 9.0;
        double r421767 = r421761 * r421766;
        double r421768 = r421765 / r421767;
        double r421769 = r421764 + r421768;
        double r421770 = r421769 - r421765;
        double r421771 = r421763 * r421770;
        return r421771;
}
↓
double f(double x, double y) {
        double r421772 = 3.0;
        double r421773 = cbrt(r421772);
        double r421774 = r421773 * r421773;
        double r421775 = sqrt(r421773);
        double r421776 = x;
        double r421777 = sqrt(r421776);
        double r421778 = y;
        double r421779 = 1.0;
        double r421780 = r421779 / r421776;
        double r421781 = 9.0;
        double r421782 = r421780 / r421781;
        double r421783 = r421778 + r421782;
        double r421784 = r421783 - r421779;
        double r421785 = r421777 * r421784;
        double r421786 = r421775 * r421785;
        double r421787 = r421775 * r421786;
        double r421788 = r421774 * r421787;
        return r421788;
}

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

Derivation

  1. Initial program 0.4

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

  3. Applied associate-/r*0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied add-cube-cbrt0.4

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

  8. Applied associate-*l*0.6

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

  10. Applied add-sqr-sqrt0.6

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

  11. Applied associate-*l*0.5

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

  12. Final simplification0.5

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

Reproduce

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

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

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