Average Error: 0.4 → 0.4
Time: 13.1s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\left(\frac{\frac{2001599834386887}{18014398509481984}}{x} + \left(y - 1\right)\right) \cdot \sqrt{x}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\left(\frac{\frac{2001599834386887}{18014398509481984}}{x} + \left(y - 1\right)\right) \cdot \sqrt{x}\right)
double f(double x, double y) {
        double r283800 = 3.0;
        double r283801 = x;
        double r283802 = sqrt(r283801);
        double r283803 = r283800 * r283802;
        double r283804 = y;
        double r283805 = 1.0;
        double r283806 = 9.0;
        double r283807 = r283801 * r283806;
        double r283808 = r283805 / r283807;
        double r283809 = r283804 + r283808;
        double r283810 = r283809 - r283805;
        double r283811 = r283803 * r283810;
        return r283811;
}
↓
double f(double x, double y) {
        double r283812 = 3.0;
        double r283813 = 2001599834386887.0;
        double r283814 = 18014398509481984.0;
        double r283815 = r283813 / r283814;
        double r283816 = x;
        double r283817 = r283815 / r283816;
        double r283818 = y;
        double r283819 = 1.0;
        double r283820 = r283818 - r283819;
        double r283821 = r283817 + r283820;
        double r283822 = sqrt(r283816);
        double r283823 = r283821 * r283822;
        double r283824 = r283812 * r283823;
        return r283824;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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. Taylor expanded around 0 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + 0.1111111111111111049432054187491303309798 \cdot \frac{1}{x}\right)} - 1\right)\]

  3. Simplified0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right)} - 1\right)\]
  4. Using strategy rm

  5. Applied pow10.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right) - 1\right)}^{1}}\]

  6. Applied pow10.4

    \[\leadsto \left(3 \cdot \color{blue}{{\left(\sqrt{x}\right)}^{1}}\right) \cdot {\left(\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  7. Applied pow10.4

    \[\leadsto \left(\color{blue}{{3}^{1}} \cdot {\left(\sqrt{x}\right)}^{1}\right) \cdot {\left(\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  8. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(3 \cdot \sqrt{x}\right)}^{1}} \cdot {\left(\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  9. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{2001599834386887}{18014398509481984} \cdot \frac{1}{x}\right) - 1\right)\right)}^{1}}\]

  10. Simplified0.4

    \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(\frac{\frac{2001599834386887}{18014398509481984}}{x} + \left(y - 1\right)\right)\right) \cdot \sqrt{x}\right)}}^{1}\]
  11. Using strategy rm

  12. Applied associate-*l*0.4

    \[\leadsto {\color{blue}{\left(3 \cdot \left(\left(\frac{\frac{2001599834386887}{18014398509481984}}{x} + \left(y - 1\right)\right) \cdot \sqrt{x}\right)\right)}}^{1}\]

  13. Final simplification0.4

    \[\leadsto 3 \cdot \left(\left(\frac{\frac{2001599834386887}{18014398509481984}}{x} + \left(y - 1\right)\right) \cdot \sqrt{x}\right)\]

Reproduce

herbie shell --seed 2019303 
(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)))