Average Error: 0.2 → 0.2
Time: 7.3s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
↓
\[\left|\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]
\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|\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\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
  (*
   (fma
    0.047619047619047616
    (pow (fabs x) 7.0)
    (fma
     (fabs x)
     2.0
     (fma 0.6666666666666666 (pow (fabs x) 3.0) (* 0.2 (pow (fabs x) 5.0)))))
   (sqrt (/ 1.0 PI)))))
double code(double x) {
	return fabs((1.0 / sqrt((double) M_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)))));
}
↓
double code(double x) {
	return fabs(fma(0.047619047619047616, pow(fabs(x), 7.0), fma(fabs(x), 2.0, fma(0.6666666666666666, pow(fabs(x), 3.0), (0.2 * pow(fabs(x), 5.0))))) * sqrt(1.0 / ((double) M_PI)));
}

Error

Bits error versus x

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. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.2

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

  4. Applied *-un-lft-identity_binary640.2

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

  5. Applied times-frac_binary640.2

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

  6. Applied associate-*l*_binary640.5

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

  7. Simplified0.3

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

  8. Taylor expanded in x around 0 0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2021211 
(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)))))))