Average Error: 0.2 → 0.2
Time: 20.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}
double f(double x, double y) {
        double r15107156 = 1.0;
        double r15107157 = x;
        double r15107158 = 9.0;
        double r15107159 = r15107157 * r15107158;
        double r15107160 = r15107156 / r15107159;
        double r15107161 = r15107156 - r15107160;
        double r15107162 = y;
        double r15107163 = 3.0;
        double r15107164 = sqrt(r15107157);
        double r15107165 = r15107163 * r15107164;
        double r15107166 = r15107162 / r15107165;
        double r15107167 = r15107161 - r15107166;
        return r15107167;
}
↓
double f(double x, double y) {
        double r15107168 = 1.0;
        double r15107169 = 0.1111111111111111;
        double r15107170 = x;
        double r15107171 = r15107169 / r15107170;
        double r15107172 = r15107168 - r15107171;
        double r15107173 = y;
        double r15107174 = sqrt(r15107170);
        double r15107175 = 3.0;
        double r15107176 = r15107174 * r15107175;
        double r15107177 = r15107173 / r15107176;
        double r15107178 = r15107172 - r15107177;
        return r15107178;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  3. Final simplification0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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)))))