Average Error: 0.4 → 0.4
Time: 4.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) - 1\right)\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) - 1\right)\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r391607 = 3.0;
        double r391608 = x;
        double r391609 = sqrt(r391608);
        double r391610 = r391607 * r391609;
        double r391611 = y;
        double r391612 = 1.0;
        double r391613 = 9.0;
        double r391614 = r391608 * r391613;
        double r391615 = r391612 / r391614;
        double r391616 = r391611 + r391615;
        double r391617 = r391616 - r391612;
        double r391618 = r391610 * r391617;
        return r391618;
}
↓
double f(double x, double y) {
        double r391619 = 3.0;
        double r391620 = 1.0;
        double r391621 = x;
        double r391622 = 9.0;
        double r391623 = r391621 * r391622;
        double r391624 = r391620 / r391623;
        double r391625 = y;
        double r391626 = r391624 + r391625;
        double r391627 = r391626 - r391620;
        double r391628 = r391619 * r391627;
        double r391629 = sqrt(r391621);
        double r391630 = r391628 * r391629;
        return r391630;
}

Error

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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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  5. Applied pow10.4

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

  6. Applied pow10.4

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

  7. Applied pow10.4

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

  8. Applied pow-prod-down0.4

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

  9. Applied pow-prod-down0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020081 
(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)))