Average Error: 0.2 → 0.3
Time: 16.6s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{\sqrt{x}} \cdot \frac{1}{3}
double f(double x, double y) {
        double r320800 = 1.0;
        double r320801 = x;
        double r320802 = 9.0;
        double r320803 = r320801 * r320802;
        double r320804 = r320800 / r320803;
        double r320805 = r320800 - r320804;
        double r320806 = y;
        double r320807 = 3.0;
        double r320808 = sqrt(r320801);
        double r320809 = r320807 * r320808;
        double r320810 = r320806 / r320809;
        double r320811 = r320805 - r320810;
        return r320811;
}
↓
double f(double x, double y) {
        double r320812 = 1.0;
        double r320813 = 9.0;
        double r320814 = x;
        double r320815 = r320813 * r320814;
        double r320816 = r320812 / r320815;
        double r320817 = r320812 - r320816;
        double r320818 = y;
        double r320819 = sqrt(r320814);
        double r320820 = r320818 / r320819;
        double r320821 = 1.0;
        double r320822 = 3.0;
        double r320823 = r320821 / r320822;
        double r320824 = r320820 * r320823;
        double r320825 = r320817 - r320824;
        return r320825;
}

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

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019196 
(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)))))