Average Error: 0.4 → 0.6
Time: 13.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(3 \cdot {x}^{\frac{1}{4}}\right) \cdot \sqrt{\sqrt{x}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(3 \cdot {x}^{\frac{1}{4}}\right) \cdot \sqrt{\sqrt{x}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
double f(double x, double y) {
        double r347837 = 3.0;
        double r347838 = x;
        double r347839 = sqrt(r347838);
        double r347840 = r347837 * r347839;
        double r347841 = y;
        double r347842 = 1.0;
        double r347843 = 9.0;
        double r347844 = r347838 * r347843;
        double r347845 = r347842 / r347844;
        double r347846 = r347841 + r347845;
        double r347847 = r347846 - r347842;
        double r347848 = r347840 * r347847;
        return r347848;
}
↓
double f(double x, double y) {
        double r347849 = 3.0;
        double r347850 = x;
        double r347851 = 0.25;
        double r347852 = pow(r347850, r347851);
        double r347853 = r347849 * r347852;
        double r347854 = sqrt(r347850);
        double r347855 = sqrt(r347854);
        double r347856 = r347853 * r347855;
        double r347857 = y;
        double r347858 = 1.0;
        double r347859 = 9.0;
        double r347860 = r347850 * r347859;
        double r347861 = r347858 / r347860;
        double r347862 = r347857 + r347861;
        double r347863 = r347862 - r347858;
        double r347864 = r347856 * r347863;
        return r347864;
}

Error

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Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.6
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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 add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.7

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

  5. Applied associate-*r*0.6

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

  6. Taylor expanded around 0 0.6

    \[\leadsto \left(\color{blue}{\left(3 \cdot {x}^{\frac{1}{4}}\right)} \cdot \sqrt{\sqrt{x}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

  7. Final simplification0.6

    \[\leadsto \left(\left(3 \cdot {x}^{\frac{1}{4}}\right) \cdot \sqrt{\sqrt{x}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))