Average Error: 0.2 → 0.2
Time: 7.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[1 - \left(\frac{\frac{y}{\sqrt{x}}}{3} + \frac{\frac{1}{x}}{9}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
1 - \left(\frac{\frac{y}{\sqrt{x}}}{3} + \frac{\frac{1}{x}}{9}\right)
double f(double x, double y) {
        double r478229 = 1.0;
        double r478230 = x;
        double r478231 = 9.0;
        double r478232 = r478230 * r478231;
        double r478233 = r478229 / r478232;
        double r478234 = r478229 - r478233;
        double r478235 = y;
        double r478236 = 3.0;
        double r478237 = sqrt(r478230);
        double r478238 = r478236 * r478237;
        double r478239 = r478235 / r478238;
        double r478240 = r478234 - r478239;
        return r478240;
}
↓
double f(double x, double y) {
        double r478241 = 1.0;
        double r478242 = y;
        double r478243 = x;
        double r478244 = sqrt(r478243);
        double r478245 = r478242 / r478244;
        double r478246 = 3.0;
        double r478247 = r478245 / r478246;
        double r478248 = r478241 / r478243;
        double r478249 = 9.0;
        double r478250 = r478248 / r478249;
        double r478251 = r478247 + r478250;
        double r478252 = r478241 - r478251;
        return r478252;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

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. Using strategy rm

  6. Applied associate-/r*0.3

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

  8. Applied sub-neg0.3

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

  9. Applied associate--l+0.3

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 1978988140 
(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)))))