Average Error: 0.2 → 0.1
Time: 8.3s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}} \cdot \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}}\right)\right)\right|\]
\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|\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)}^{3.5}}{\sqrt{21}} \cdot \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}}\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) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((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) ((((double) pow(((double) fabs(x)), 3.5)) / ((double) sqrt(21.0))) * (((double) pow(((double) fabs(x)), 3.5)) / ((double) sqrt(21.0)))))))))))))));
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied sqr-pow0.2

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \color{blue}{\frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}}} \cdot \frac{{\left(\left|x\right|\right)}^{\left(\frac{7}{2}\right)}}{\sqrt{21}}\right)\right)\right|\]

  8. Simplified0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}} \cdot \color{blue}{\frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}}}\right)\right)\right|\]

  9. Final simplification0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left(\left|x\right| + \frac{{\left(\left|x\right|\right)}^{3}}{3}\right) + 1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{5}}{5} + \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}} \cdot \frac{{\left(\left|x\right|\right)}^{3.5}}{\sqrt{21}}\right)\right)\right|\]

Reproduce

herbie shell --seed 2020196 
(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)))))))