Average Error: 0.2 → 0.3
Time: 19.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}
double f(double x, double y) {
        double r16614764 = 1.0;
        double r16614765 = x;
        double r16614766 = 9.0;
        double r16614767 = r16614765 * r16614766;
        double r16614768 = r16614764 / r16614767;
        double r16614769 = r16614764 - r16614768;
        double r16614770 = y;
        double r16614771 = 3.0;
        double r16614772 = sqrt(r16614765);
        double r16614773 = r16614771 * r16614772;
        double r16614774 = r16614770 / r16614773;
        double r16614775 = r16614769 - r16614774;
        return r16614775;
}
↓
double f(double x, double y) {
        double r16614776 = 1.0;
        double r16614777 = 0.1111111111111111;
        double r16614778 = x;
        double r16614779 = r16614777 / r16614778;
        double r16614780 = r16614776 - r16614779;
        double r16614781 = 1.0;
        double r16614782 = 3.0;
        double r16614783 = r16614781 / r16614782;
        double r16614784 = y;
        double r16614785 = sqrt(r16614778);
        double r16614786 = r16614784 / r16614785;
        double r16614787 = r16614783 * r16614786;
        double r16614788 = r16614780 - r16614787;
        return r16614788;
}

Error

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Bits error versus y

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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. Taylor expanded around 0 0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +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)))))