Average Error: 2.8 → 1.2
Time: 10.5s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left({\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\right)\]
\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)
↓
\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left({\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\right)
(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
 (*
  (/ 1.0 (sqrt PI))
  (*
   (pow (exp (fabs x)) (/ (fabs x) 2.0))
   (*
    (pow (exp (fabs x)) (/ (fabs x) 2.0))
    (+
     (/ 1.0 (fabs x))
     (+
      (* (/ 1.0 2.0) (pow (/ 1.0 (fabs x)) 3.0))
      (+
       (* (pow (/ 1.0 (fabs x)) 5.0) (/ 3.0 4.0))
       (* (/ 15.0 8.0) (/ 1.0 (pow (fabs x) 7.0))))))))))
double code(double x) {
	return ((double) (((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) exp(((double) (((double) fabs(x)) * ((double) fabs(x)))))))) * ((double) (((double) (((double) ((1.0 / ((double) fabs(x))) + ((double) ((1.0 / 2.0) * ((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((3.0 / 4.0) * ((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((15.0 / 8.0) * ((double) (((double) (((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x)))))))))));
}
↓
double code(double x) {
	return ((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) (((double) pow(((double) exp(((double) fabs(x)))), (((double) fabs(x)) / 2.0))) * ((double) (((double) pow(((double) exp(((double) fabs(x)))), (((double) fabs(x)) / 2.0))) * ((double) ((1.0 / ((double) fabs(x))) + ((double) (((double) ((1.0 / 2.0) * ((double) pow((1.0 / ((double) fabs(x))), 3.0)))) + ((double) (((double) (((double) pow((1.0 / ((double) fabs(x))), 5.0)) * (3.0 / 4.0))) + ((double) ((15.0 / 8.0) * (1.0 / ((double) pow(((double) fabs(x)), 7.0)))))))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 2.8 bits

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

  2. SimplifiedError: 1.4 bits

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)\right)\right)}\]

  3. Taylor expanded around 0 Error: 1.2 bits

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{7}}}\right)\right)\right)\right)\]
  4. Using strategy rm

  5. Applied sqr-powError: 1.3 bits

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\]

  6. Applied associate-*l*Error: 1.2 bits

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\right)}\]

  7. SimplifiedError: 1.2 bits

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \color{blue}{\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left({\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)}\right)\]

  8. Final simplificationError: 1.2 bits

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left({\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020204 
(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)))))))