Average Error: 0.2 → 0.3
Time: 1.5m
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(\frac{1}{3} + \frac{-1}{3}\right) \cdot \frac{y}{\sqrt{x}} + \left(\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}\right) - \frac{\frac{y}{\sqrt{x}}}{3}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(\frac{1}{3} + \frac{-1}{3}\right) \cdot \frac{y}{\sqrt{x}} + \left(\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}\right) - \frac{\frac{y}{\sqrt{x}}}{3}\right)
double f(double x, double y) {
        double r389747 = 1.0;
        double r389748 = x;
        double r389749 = 9.0;
        double r389750 = r389748 * r389749;
        double r389751 = r389747 / r389750;
        double r389752 = r389747 - r389751;
        double r389753 = y;
        double r389754 = 3.0;
        double r389755 = sqrt(r389748);
        double r389756 = r389754 * r389755;
        double r389757 = r389753 / r389756;
        double r389758 = r389752 - r389757;
        return r389758;
}
↓
double f(double x, double y) {
        double r389759 = 1.0;
        double r389760 = 3.0;
        double r389761 = r389759 / r389760;
        double r389762 = -1.0;
        double r389763 = r389762 / r389760;
        double r389764 = r389761 + r389763;
        double r389765 = y;
        double r389766 = x;
        double r389767 = sqrt(r389766);
        double r389768 = r389765 / r389767;
        double r389769 = r389764 * r389768;
        double r389770 = 1.0;
        double r389771 = r389770 / r389766;
        double r389772 = 9.0;
        double r389773 = sqrt(r389772);
        double r389774 = r389771 / r389773;
        double r389775 = r389774 / r389773;
        double r389776 = r389770 - r389775;
        double r389777 = r389768 / r389760;
        double r389778 = r389776 - r389777;
        double r389779 = r389769 + r389778;
        return r389779;
}

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 *-un-lft-identity0.2

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

  4. Applied times-frac0.3

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

  5. Applied add-sqr-sqrt29.9

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

  6. Applied prod-diff29.9

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

  7. Simplified0.2

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

  8. Simplified0.2

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

  10. Applied associate-/r*0.2

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

  12. Applied add-sqr-sqrt0.2

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

  13. Applied associate-/r*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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)))))