Average Error: 0.2 → 0.1
Time: 29.0min
Precision: binary64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{\sqrt{\pi}} \cdot \left|x\right|\right) \cdot \left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{4}}{5} + \frac{{\left(\left|x\right|\right)}^{6}}{21}\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{2} + 2\right)\right)\right|\]
\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|\left(\frac{1}{\sqrt{\pi}} \cdot \left|x\right|\right) \cdot \left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{4}}{5} + \frac{{\left(\left|x\right|\right)}^{6}}{21}\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{2} + 2\right)\right)\right|
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) (((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) fabs(x)))) * ((double) (((double) (1.0 * ((double) ((((double) pow(((double) fabs(x)), 4.0)) / 5.0) + (((double) pow(((double) fabs(x)), 6.0)) / 21.0))))) + ((double) (((double) ((2.0 / 3.0) * ((double) pow(((double) fabs(x)), 2.0)))) + 2.0))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21} + {\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5}\right)\right) + {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right|}\]
  3. Using strategy rm

  4. Applied pow10.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21} + {\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5}\right)\right) + {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)}^{1}}\right|\]

  5. Applied pow10.2

    \[\leadsto \left|\color{blue}{{\left(\frac{1}{\sqrt{\pi}}\right)}^{1}} \cdot {\left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21} + {\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5}\right)\right) + {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)}^{1}\right|\]

  6. Applied pow-prod-down0.2

    \[\leadsto \left|\color{blue}{{\left(\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21} + {\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5}\right)\right) + {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right)}^{1}}\right|\]

  7. Simplified0.1

    \[\leadsto \left|{\color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \left|x\right|\right) \cdot \left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{4}}{5} + \frac{{\left(\left|x\right|\right)}^{6}}{21}\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{2} + 2\right)\right)\right)}}^{1}\right|\]

  8. Final simplification0.1

    \[\leadsto \left|\left(\frac{1}{\sqrt{\pi}} \cdot \left|x\right|\right) \cdot \left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{4}}{5} + \frac{{\left(\left|x\right|\right)}^{6}}{21}\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{2} + 2\right)\right)\right|\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :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)))))))