Average Error: 0.2 → 0.3
Time: 16.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r27107037 = 1.0;
        double r27107038 = x;
        double r27107039 = 9.0;
        double r27107040 = r27107038 * r27107039;
        double r27107041 = r27107037 / r27107040;
        double r27107042 = r27107037 - r27107041;
        double r27107043 = y;
        double r27107044 = 3.0;
        double r27107045 = sqrt(r27107038);
        double r27107046 = r27107044 * r27107045;
        double r27107047 = r27107043 / r27107046;
        double r27107048 = r27107042 - r27107047;
        return r27107048;
}
↓
double f(double x, double y) {
        double r27107049 = 1.0;
        double r27107050 = x;
        double r27107051 = r27107049 / r27107050;
        double r27107052 = 9.0;
        double r27107053 = r27107051 / r27107052;
        double r27107054 = r27107049 - r27107053;
        double r27107055 = y;
        double r27107056 = 1.0;
        double r27107057 = 3.0;
        double r27107058 = sqrt(r27107050);
        double r27107059 = r27107057 * r27107058;
        double r27107060 = r27107056 / r27107059;
        double r27107061 = r27107055 * r27107060;
        double r27107062 = r27107054 - r27107061;
        return r27107062;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.3
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*0.3

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

  5. Applied div-inv0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 
(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)))))