Average Error: 0.4 → 0.4
Time: 42.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(\left(\frac{\frac{1}{x}}{9} - 1\right) + y\right) \cdot \sqrt{x}\right) \cdot 3 + \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(-1\right) + 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(\left(\frac{\frac{1}{x}}{9} - 1\right) + y\right) \cdot \sqrt{x}\right) \cdot 3 + \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(-1\right) + 1\right)
double f(double x, double y) {
        double r379988 = 3.0;
        double r379989 = x;
        double r379990 = sqrt(r379989);
        double r379991 = r379988 * r379990;
        double r379992 = y;
        double r379993 = 1.0;
        double r379994 = 9.0;
        double r379995 = r379989 * r379994;
        double r379996 = r379993 / r379995;
        double r379997 = r379992 + r379996;
        double r379998 = r379997 - r379993;
        double r379999 = r379991 * r379998;
        return r379999;
}
↓
double f(double x, double y) {
        double r380000 = 1.0;
        double r380001 = x;
        double r380002 = r380000 / r380001;
        double r380003 = 9.0;
        double r380004 = r380002 / r380003;
        double r380005 = r380004 - r380000;
        double r380006 = y;
        double r380007 = r380005 + r380006;
        double r380008 = sqrt(r380001);
        double r380009 = r380007 * r380008;
        double r380010 = 3.0;
        double r380011 = r380009 * r380010;
        double r380012 = r380010 * r380008;
        double r380013 = -r380000;
        double r380014 = r380013 + r380000;
        double r380015 = r380012 * r380014;
        double r380016 = r380011 + r380015;
        return r380016;
}

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 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 add-cube-cbrt0.4

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

  6. Applied add-sqr-sqrt15.2

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

  7. Applied prod-diff15.2

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

  8. Applied distribute-lft-in15.2

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))