Average Error: 0.4 → 0.4
Time: 21.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \sqrt{x} + \left(\left(1 + \left(-1\right)\right) \cdot 3\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \sqrt{x} + \left(\left(1 + \left(-1\right)\right) \cdot 3\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r336944 = 3.0;
        double r336945 = x;
        double r336946 = sqrt(r336945);
        double r336947 = r336944 * r336946;
        double r336948 = y;
        double r336949 = 1.0;
        double r336950 = 9.0;
        double r336951 = r336945 * r336950;
        double r336952 = r336949 / r336951;
        double r336953 = r336948 + r336952;
        double r336954 = r336953 - r336949;
        double r336955 = r336947 * r336954;
        return r336955;
}
↓
double f(double x, double y) {
        double r336956 = y;
        double r336957 = 1.0;
        double r336958 = x;
        double r336959 = r336957 / r336958;
        double r336960 = 9.0;
        double r336961 = r336959 / r336960;
        double r336962 = r336956 + r336961;
        double r336963 = r336962 - r336957;
        double r336964 = 3.0;
        double r336965 = r336963 * r336964;
        double r336966 = sqrt(r336958);
        double r336967 = r336965 * r336966;
        double r336968 = -r336957;
        double r336969 = r336957 + r336968;
        double r336970 = r336969 * r336964;
        double r336971 = r336970 * r336966;
        double r336972 = r336967 + r336971;
        return r336972;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-/r*0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied add-sqr-sqrt15.2

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

  7. Applied prod-diff15.2

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

  8. Applied distribute-rgt-in15.2

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +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)))