Average Error: 2.8 → 1.2
Time: 11.6s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
↓
\[\begin{array}{l} t_0 := {\left(e^{x}\right)}^{x}\\ \frac{\frac{1}{x}}{\frac{\sqrt{\pi}}{t_0}} + \mathsf{fma}\left(1.875, {x}^{-6}, {x}^{-2} \cdot \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right)\right) \cdot \left(t_0 \cdot \frac{{\pi}^{-0.5}}{x}\right) \end{array} \]
\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)
↓
\begin{array}{l}
t_0 := {\left(e^{x}\right)}^{x}\\
\frac{\frac{1}{x}}{\frac{\sqrt{\pi}}{t_0}} + \mathsf{fma}\left(1.875, {x}^{-6}, {x}^{-2} \cdot \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right)\right) \cdot \left(t_0 \cdot \frac{{\pi}^{-0.5}}{x}\right)
\end{array}

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
  (+
   (+
    (+
     (/ 1.0 (fabs x))
     (*
      (/ 1.0 2.0)
      (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))
    (*
     (/ 3.0 4.0)
     (*
      (*
       (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))))
   (*
    (/ 15.0 8.0)
    (*
     (*
      (*
       (*
        (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
        (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))
     (/ 1.0 (fabs x)))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) x)))
   (+
    (/ (/ 1.0 x) (/ (sqrt PI) t_0))
    (*
     (fma 1.875 (pow x -6.0) (* (pow x -2.0) (fma 0.75 (pow x -2.0) 0.5)))
     (* t_0 (/ (pow PI -0.5) x))))))
double code(double x) {
	return ((1.0 / sqrt((double) M_PI)) * exp(fabs(x) * fabs(x))) * ((((1.0 / fabs(x)) + ((1.0 / 2.0) * (((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((3.0 / 4.0) * (((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((15.0 / 8.0) * (((((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x)))));
}
↓
double code(double x) {
	double t_0 = pow(exp(x), x);
	return ((1.0 / x) / (sqrt((double) M_PI) / t_0)) + (fma(1.875, pow(x, -6.0), (pow(x, -2.0) * fma(0.75, pow(x, -2.0), 0.5))) * (t_0 * (pow(((double) M_PI), -0.5) / x)));
}

Error

Bits error versus x

Derivation

  1. Initial program 2.8

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

  2. Simplified2.7

    \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

  3. Applied egg1.3

    \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

  4. Applied egg1.3

    \[\leadsto \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}}{\left|x\right|} + \mathsf{fma}\left(1.875, {x}^{-6}, \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right) \cdot {x}^{-2}\right) \cdot \frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}}{\left|x\right|}} \]

  5. Applied egg1.2

    \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}}{\left|x\right|} + \mathsf{fma}\left(1.875, {x}^{-6}, \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right) \cdot {x}^{-2}\right) \cdot \color{blue}{\left(\frac{{\left(e^{x}\right)}^{x}}{1} \cdot \frac{{\pi}^{-0.5}}{x}\right)} \]

  6. Applied egg1.2

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{\sqrt{\pi}}{{\left(e^{x}\right)}^{x}}}} + \mathsf{fma}\left(1.875, {x}^{-6}, \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right) \cdot {x}^{-2}\right) \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{1} \cdot \frac{{\pi}^{-0.5}}{x}\right) \]

  7. Final simplification1.2

    \[\leadsto \frac{\frac{1}{x}}{\frac{\sqrt{\pi}}{{\left(e^{x}\right)}^{x}}} + \mathsf{fma}\left(1.875, {x}^{-6}, {x}^{-2} \cdot \mathsf{fma}\left(0.75, {x}^{-2}, 0.5\right)\right) \cdot \left({\left(e^{x}\right)}^{x} \cdot \frac{{\pi}^{-0.5}}{x}\right) \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))