Average Error: 0.2 → 0.2
Time: 9.6s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r469216 = 1.0;
        double r469217 = x;
        double r469218 = 9.0;
        double r469219 = r469217 * r469218;
        double r469220 = r469216 / r469219;
        double r469221 = r469216 - r469220;
        double r469222 = y;
        double r469223 = 3.0;
        double r469224 = sqrt(r469217);
        double r469225 = r469223 * r469224;
        double r469226 = r469222 / r469225;
        double r469227 = r469221 - r469226;
        return r469227;
}
↓
double f(double x, double y) {
        double r469228 = 1.0;
        double r469229 = 0.1111111111111111;
        double r469230 = x;
        double r469231 = r469229 / r469230;
        double r469232 = r469228 - r469231;
        double r469233 = y;
        double r469234 = 3.0;
        double r469235 = r469233 / r469234;
        double r469236 = sqrt(r469230);
        double r469237 = r469235 / r469236;
        double r469238 = r469232 - r469237;
        return r469238;
}

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 associate-/r*0.2

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020035 +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)))))