Average Error: 0.2 → 0.3
Time: 10.6s
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 r419129 = 1.0;
        double r419130 = x;
        double r419131 = 9.0;
        double r419132 = r419130 * r419131;
        double r419133 = r419129 / r419132;
        double r419134 = r419129 - r419133;
        double r419135 = y;
        double r419136 = 3.0;
        double r419137 = sqrt(r419130);
        double r419138 = r419136 * r419137;
        double r419139 = r419135 / r419138;
        double r419140 = r419134 - r419139;
        return r419140;
}
↓
double f(double x, double y) {
        double r419141 = 1.0;
        double r419142 = x;
        double r419143 = r419141 / r419142;
        double r419144 = 9.0;
        double r419145 = r419143 / r419144;
        double r419146 = r419141 - r419145;
        double r419147 = 1.0;
        double r419148 = 3.0;
        double r419149 = r419147 / r419148;
        double r419150 = y;
        double r419151 = sqrt(r419142);
        double r419152 = r419150 / r419151;
        double r419153 = r419149 * r419152;
        double r419154 = r419146 - r419153;
        return r419154;
}

Error

Bits error versus x

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 2020045 
(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)))))