Average Error: 0.4 → 0.5
Time: 4.5s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(y + \frac{\frac{1}{x}}{9}\right)\right) \cdot \sqrt{x} + 3 \cdot \left(\left(-1\right) \cdot \sqrt{x}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(y + \frac{\frac{1}{x}}{9}\right)\right) \cdot \sqrt{x} + 3 \cdot \left(\left(-1\right) \cdot \sqrt{x}\right)
double f(double x, double y) {
        double r325722 = 3.0;
        double r325723 = x;
        double r325724 = sqrt(r325723);
        double r325725 = r325722 * r325724;
        double r325726 = y;
        double r325727 = 1.0;
        double r325728 = 9.0;
        double r325729 = r325723 * r325728;
        double r325730 = r325727 / r325729;
        double r325731 = r325726 + r325730;
        double r325732 = r325731 - r325727;
        double r325733 = r325725 * r325732;
        return r325733;
}
↓
double f(double x, double y) {
        double r325734 = 3.0;
        double r325735 = y;
        double r325736 = 1.0;
        double r325737 = x;
        double r325738 = r325736 / r325737;
        double r325739 = 9.0;
        double r325740 = r325738 / r325739;
        double r325741 = r325735 + r325740;
        double r325742 = r325734 * r325741;
        double r325743 = sqrt(r325737);
        double r325744 = r325742 * r325743;
        double r325745 = -r325736;
        double r325746 = r325745 * r325743;
        double r325747 = r325734 * r325746;
        double r325748 = r325744 + r325747;
        return r325748;
}

Error

Bits error versus x

Bits error versus y

Try it out

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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-*l*0.4

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

  5. Applied sub-neg0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied associate-*r*0.5

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

  13. Applied associate-/r*0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 
(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)))