Average Error: 0.1 → 0.1
Time: 5.1s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666 + \left({\left(\left|x\right|\right)}^{5} \cdot 0.2 + \left(\left|x\right| \cdot 2 + {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right)\right)\right)\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
↓
\left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666 + \left({\left(\left|x\right|\right)}^{5} \cdot 0.2 + \left(\left|x\right| \cdot 2 + {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right)\right)\right)\right|
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x))))
     (*
      (/ 1.0 5.0)
      (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))))
    (*
     (/ 1.0 21.0)
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x)))))))
↓
(FPCore (x)
 :precision binary64
 (fabs
  (*
   1.0
   (*
    (sqrt (/ 1.0 PI))
    (+
     (* (pow (fabs x) 3.0) 0.6666666666666666)
     (+
      (* (pow (fabs x) 5.0) 0.2)
      (+ (* (fabs x) 2.0) (* (pow (fabs x) 7.0) 0.047619047619047616))))))))
double code(double x) {
	return ((double) fabs(((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) (((double) (((double) (((double) (2.0 * ((double) fabs(x)))) + ((double) ((2.0 / 3.0) * ((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))))))) + ((double) ((1.0 / 5.0) * ((double) (((double) (((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))))))) + ((double) ((1.0 / 21.0) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x))))))))))));
}
↓
double code(double x) {
	return ((double) fabs(((double) (1.0 * ((double) (((double) sqrt((1.0 / ((double) M_PI)))) * ((double) (((double) (((double) pow(((double) fabs(x)), 3.0)) * 0.6666666666666666)) + ((double) (((double) (((double) pow(((double) fabs(x)), 5.0)) * 0.2)) + ((double) (((double) (((double) fabs(x)) * 2.0)) + ((double) (((double) pow(((double) fabs(x)), 7.0)) * 0.047619047619047616))))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 0.1 bits

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. SimplifiedError: 0.1 bits

    \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \frac{{\left(\left|x\right|\right)}^{7}}{21}\right)\right)\right|}\]

  3. Taylor expanded around 0 Error: 0.1 bits

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.2 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]

  4. SimplifiedError: 0.1 bits

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666 + \left({\left(\left|x\right|\right)}^{5} \cdot 0.2 + \left(2 \cdot \left|x\right| + {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right)\right)\right)}\right|\]

  5. Final simplificationError: 0.1 bits

    \[\leadsto \left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666 + \left({\left(\left|x\right|\right)}^{5} \cdot 0.2 + \left(\left|x\right| \cdot 2 + {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))