Average Error: 0.2 → 0.3
Time: 16.8s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{y}{\sqrt{x}}}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{y}{\sqrt{x}}}{3}
double f(double x, double y) {
        double r441798 = 1.0;
        double r441799 = x;
        double r441800 = 9.0;
        double r441801 = r441799 * r441800;
        double r441802 = r441798 / r441801;
        double r441803 = r441798 - r441802;
        double r441804 = y;
        double r441805 = 3.0;
        double r441806 = sqrt(r441799);
        double r441807 = r441805 * r441806;
        double r441808 = r441804 / r441807;
        double r441809 = r441803 - r441808;
        return r441809;
}
↓
double f(double x, double y) {
        double r441810 = 1.0;
        double r441811 = 9.0;
        double r441812 = r441810 / r441811;
        double r441813 = x;
        double r441814 = r441812 / r441813;
        double r441815 = r441810 - r441814;
        double r441816 = y;
        double r441817 = sqrt(r441813);
        double r441818 = r441816 / r441817;
        double r441819 = 3.0;
        double r441820 = r441818 / r441819;
        double r441821 = r441815 - r441820;
        return r441821;
}

Error

Bits error versus x

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

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

  4. Applied *-un-lft-identity0.2

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

  5. Applied distribute-lft-out--0.2

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))