Average Error: 0.2 → 0.3
Time: 4.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111}{x}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111}{x}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r425959 = 1.0;
        double r425960 = x;
        double r425961 = 9.0;
        double r425962 = r425960 * r425961;
        double r425963 = r425959 / r425962;
        double r425964 = r425959 - r425963;
        double r425965 = y;
        double r425966 = 3.0;
        double r425967 = sqrt(r425960);
        double r425968 = r425966 * r425967;
        double r425969 = r425965 / r425968;
        double r425970 = r425964 - r425969;
        return r425970;
}
↓
double f(double x, double y) {
        double r425971 = 1.0;
        double r425972 = 0.1111111111111111;
        double r425973 = x;
        double r425974 = r425972 / r425973;
        double r425975 = r425971 - r425974;
        double r425976 = y;
        double r425977 = 1.0;
        double r425978 = 3.0;
        double r425979 = sqrt(r425973);
        double r425980 = r425978 * r425979;
        double r425981 = r425977 / r425980;
        double r425982 = r425976 * r425981;
        double r425983 = r425975 - r425982;
        return r425983;
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

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 div-inv0.2

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

  4. Taylor expanded around 0 0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 +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)))))