Average Error: 0.2 → 0.3
Time: 8.4s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}
double f(double x, double y) {
        double r325170 = 1.0;
        double r325171 = x;
        double r325172 = 9.0;
        double r325173 = r325171 * r325172;
        double r325174 = r325170 / r325173;
        double r325175 = r325170 - r325174;
        double r325176 = y;
        double r325177 = 3.0;
        double r325178 = sqrt(r325171);
        double r325179 = r325177 * r325178;
        double r325180 = r325176 / r325179;
        double r325181 = r325175 - r325180;
        return r325181;
}
↓
double f(double x, double y) {
        double r325182 = 1.0;
        double r325183 = x;
        double r325184 = r325182 / r325183;
        double r325185 = 9.0;
        double r325186 = r325184 / r325185;
        double r325187 = r325182 - r325186;
        double r325188 = 1.0;
        double r325189 = 3.0;
        double r325190 = r325188 / r325189;
        double r325191 = y;
        double r325192 = sqrt(r325183);
        double r325193 = r325191 / r325192;
        double r325194 = r325190 * r325193;
        double r325195 = r325187 - r325194;
        return r325195;
}

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 associate-/r*0.2

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

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

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

  6. Applied times-frac0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020039 +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)))))