Average Error: 0.2 → 0.4
Time: 5.2s
Precision: binary64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \frac{2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \frac{{\left(\left|x\right|\right)}^{7}}{21}\right)}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right|\]
\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 \frac{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)}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\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) (1.0 * (((double) (((double) (2.0 * ((double) (((double) fabs(x)) + (((double) pow(((double) fabs(x)), 3.0)) / 3.0))))) + ((double) (1.0 * ((double) ((((double) pow(((double) fabs(x)), 5.0)) / 5.0) + (((double) pow(((double) fabs(x)), 7.0)) / 21.0))))))) / ((double) (((double) sqrt(((double) sqrt(((double) M_PI))))) * ((double) sqrt(((double) sqrt(((double) M_PI))))))))))));
}

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

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

  4. Applied div-inv0.2

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

  5. Applied associate-*l*0.2

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

  6. Simplified0.6

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

  8. Applied add-sqr-sqrt0.6

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

  9. Applied sqrt-prod0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(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)))))))