Average Error: 0.2 → 0.1
Time: 4.8s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \frac{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \]
\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|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \frac{\mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\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
  (+
   (* (sqrt (/ 1.0 PI)) (fma 2.0 x (* 0.6666666666666666 (pow x 3.0))))
   (/ (fma 0.2 (pow x 5.0) (* 0.047619047619047616 (pow x 7.0))) (sqrt 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((sqrt(1.0 / ((double) M_PI)) * fma(2.0, x, (0.6666666666666666 * pow(x, 3.0)))) + (fma(0.2, pow(x, 5.0), (0.047619047619047616 * pow(x, 7.0))) / sqrt((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. Simplified0.6

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

  3. Taylor expanded in x around 0 0.1

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

  4. Simplified0.1

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

  5. Applied sqrt-div_binary640.1

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

  6. Applied associate-*l/_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022066 
(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)))))))