Average Error: 0.4 → 0.4
Time: 3.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)
double f(double x, double y) {
        double r411023 = 3.0;
        double r411024 = x;
        double r411025 = sqrt(r411024);
        double r411026 = r411023 * r411025;
        double r411027 = y;
        double r411028 = 1.0;
        double r411029 = 9.0;
        double r411030 = r411024 * r411029;
        double r411031 = r411028 / r411030;
        double r411032 = r411027 + r411031;
        double r411033 = r411032 - r411028;
        double r411034 = r411026 * r411033;
        return r411034;
}
↓
double f(double x, double y) {
        double r411035 = 3.0;
        double r411036 = x;
        double r411037 = sqrt(r411036);
        double r411038 = y;
        double r411039 = 0.1111111111111111;
        double r411040 = r411039 / r411036;
        double r411041 = r411038 + r411040;
        double r411042 = 1.0;
        double r411043 = r411041 - r411042;
        double r411044 = r411037 * r411043;
        double r411045 = r411035 * r411044;
        return r411045;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
  2. Using strategy rm

  3. Applied associate-*l*0.4

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

  4. Taylor expanded around 0 0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))