Average Error: 2.8 → 1.2
Time: 19.5s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := {\left(e^{x}\right)}^{x}\\ t_0 \cdot \left(0.5 \cdot \frac{{e}^{\left(x \cdot x\right)}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{t_1}{\left|x\right| \cdot {x}^{4}}\right) + t_0 \cdot \left(\frac{t_1}{\left|x\right|} + 1.875 \cdot \frac{t_1}{\left|x\right| \cdot {x}^{6}}\right) \end{array} \]
\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 := \sqrt{\frac{1}{\pi}}\\
t_1 := {\left(e^{x}\right)}^{x}\\
t_0 \cdot \left(0.5 \cdot \frac{{e}^{\left(x \cdot x\right)}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{t_1}{\left|x\right| \cdot {x}^{4}}\right) + t_0 \cdot \left(\frac{t_1}{\left|x\right|} + 1.875 \cdot \frac{t_1}{\left|x\right| \cdot {x}^{6}}\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 (sqrt (/ 1.0 PI))) (t_1 (pow (exp x) x)))
   (+
    (*
     t_0
     (+
      (* 0.5 (/ (pow E (* x x)) (pow (fabs x) 3.0)))
      (* 0.75 (/ t_1 (* (fabs x) (pow x 4.0))))))
    (* t_0 (+ (/ t_1 (fabs x)) (* 1.875 (/ t_1 (* (fabs x) (pow x 6.0)))))))))
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 = sqrt(1.0 / ((double) M_PI));
	double t_1 = pow(exp(x), x);
	return (t_0 * ((0.5 * (pow(((double) M_E), (x * x)) / pow(fabs(x), 3.0))) + (0.75 * (t_1 / (fabs(x) * pow(x, 4.0)))))) + (t_0 * ((t_1 / fabs(x)) + (1.875 * (t_1 / (fabs(x) * pow(x, 6.0))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-log-exp_binary642.7

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

  4. Applied exp-to-pow_binary641.3

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

  5. Taylor expanded in x around inf 2.6

    \[\leadsto \color{blue}{\frac{e^{\log \left(e^{x}\right) \cdot x}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}} + \left(1.875 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{\log \left(e^{x}\right) \cdot x}}{\left|x\right| \cdot {x}^{6}}\right) + \left(0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{\log \left(e^{x}\right) \cdot x}}{\left|x\right| \cdot {x}^{4}}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{\log \left(e^{x}\right) \cdot x}}{\left|x\right| \cdot {x}^{2}}\right)\right)\right)} \]

  6. Simplified1.2

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{6} \cdot \left|x\right|}\right)} \]

  7. Applied *-un-lft-identity_binary641.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{{\left(e^{\color{blue}{1 \cdot x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{6} \cdot \left|x\right|}\right) \]

  8. Applied exp-prod_binary641.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{{\color{blue}{\left({\left(e^{1}\right)}^{x}\right)}}^{x}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{6} \cdot \left|x\right|}\right) \]

  9. Applied pow-pow_binary641.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{\color{blue}{{\left(e^{1}\right)}^{\left(x \cdot x\right)}}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{6} \cdot \left|x\right|}\right) \]

  10. Final simplification1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{{e}^{\left(x \cdot x\right)}}{{\left(\left|x\right|\right)}^{3}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot {x}^{4}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot {x}^{6}}\right) \]

Reproduce

herbie shell --seed 2021224 
(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)))))))