Average Error: 0.2 → 0.2
Time: 11.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\sqrt{x}}}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\sqrt{x}}}{3}
double f(double x, double y) {
        double r324198 = 1.0;
        double r324199 = x;
        double r324200 = 9.0;
        double r324201 = r324199 * r324200;
        double r324202 = r324198 / r324201;
        double r324203 = r324198 - r324202;
        double r324204 = y;
        double r324205 = 3.0;
        double r324206 = sqrt(r324199);
        double r324207 = r324205 * r324206;
        double r324208 = r324204 / r324207;
        double r324209 = r324203 - r324208;
        return r324209;
}
↓
double f(double x, double y) {
        double r324210 = 1.0;
        double r324211 = x;
        double r324212 = 9.0;
        double r324213 = r324211 * r324212;
        double r324214 = r324210 / r324213;
        double r324215 = r324210 - r324214;
        double r324216 = y;
        double r324217 = sqrt(r324211);
        double r324218 = r324216 / r324217;
        double r324219 = 3.0;
        double r324220 = r324218 / r324219;
        double r324221 = r324215 - r324220;
        return r324221;
}

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 div-inv0.2

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019294 
(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)))))