Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.2 → 0.1

Time: 7.0s

Precision: binary64

\[x \leq 0.5\]

Math

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

↓

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left|x\right| \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)\right| \]

FPCore

(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
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* 0.6666666666666666 (pow (fabs x) 3.0)))
     (* 0.2 (* (fabs x) (* (fabs x) (* (fabs x) (* x x))))))
    (* (fabs x) (* 0.047619047619047616 (pow (fabs x) 6.0)))))))

C

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((1.0 / sqrt((double) M_PI)) * ((((2.0 * fabs(x)) + (0.6666666666666666 * pow(fabs(x), 3.0))) + (0.2 * (fabs(x) * (fabs(x) * (fabs(x) * (x * x)))))) + (fabs(x) * (0.047619047619047616 * pow(fabs(x), 6.0)))));
}

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. Applied associate-*r*_binary64 0.1

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

3. Simplified 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)} \cdot \left|x\right|\right)\right| \]

4. Applied pow1_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{{\left(\left|x\right|\right)}^{1}}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

5. Applied pow1_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \color{blue}{{\left(\left|x\right|\right)}^{1}}\right) \cdot {\left(\left|x\right|\right)}^{1}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

6. Applied pow1_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\color{blue}{{\left(\left|x\right|\right)}^{1}} \cdot {\left(\left|x\right|\right)}^{1}\right) \cdot {\left(\left|x\right|\right)}^{1}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

7. Applied pow-prod-down_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{1}} \cdot {\left(\left|x\right|\right)}^{1}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

8. Applied pow-prod-down_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \color{blue}{{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}^{1}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

9. Applied pow1_binary64 0.1

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{{\left(\frac{2}{3}\right)}^{1}} \cdot {\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}^{1}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

10. Applied pow-prod-down_binary64 0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{{\left(\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}^{1}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

11. Simplified 0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + {\color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)}}^{1}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right| \]

12. Final simplification 0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left|x\right| \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)\right| \]

Reproduce

herbie shell --seed 2022087 
(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)))))))