Average Error: 0.1 → 0.1
Time: 8.3s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) + 0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right|\]
\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|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) + 0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\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))
    (*
     (fabs x)
     (+ 2.0 (+ (* 0.6666666666666666 (* x x)) (* 0.2 (pow x 4.0))))))
   (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (* (fabs x) (pow x 6.0)))))))
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)) * (fabs(x) * (2.0 + ((0.6666666666666666 * (x * x)) + (0.2 * pow(x, 4.0)))))) + (0.047619047619047616 * (sqrt(1.0 / ((double) M_PI)) * (fabs(x) * pow(x, 6.0)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right|}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary64_35100.1

    \[\leadsto \left|\left|x\right| \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\right|\]

  5. Applied *-un-lft-identity_binary64_34880.1

    \[\leadsto \left|\left|x\right| \cdot \frac{\color{blue}{1 \cdot \left(0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right|\]

  6. Applied times-frac_binary64_34940.1

    \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\sqrt{\pi}}}\right)}\right|\]

  7. Applied associate-*r*_binary64_34280.4

    \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\sqrt{\pi}}}\right) \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\sqrt{\pi}}}}\right|\]

  8. Simplified0.3

    \[\leadsto \left|\color{blue}{\frac{\left|x\right|}{\sqrt{\sqrt{\pi}}}} \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\sqrt{\pi}}}\right|\]
  9. Using strategy rm

  10. Applied div-inv_binary64_34850.3

    \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\sqrt{\pi}}} \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) \cdot \frac{1}{\sqrt{\sqrt{\pi}}}\right)}\right|\]

  11. Applied associate-*r*_binary64_34280.3

    \[\leadsto \left|\color{blue}{\left(\frac{\left|x\right|}{\sqrt{\sqrt{\pi}}} \cdot \left(0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\sqrt{\pi}}}}\right|\]

  12. Simplified0.4

    \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\sqrt{\pi}}}\right)} \cdot \frac{1}{\sqrt{\sqrt{\pi}}}\right|\]

  13. Taylor expanded around 0 0.1

    \[\leadsto \left|\color{blue}{0.2 \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + \left(2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) + \left(0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right|\]

  14. Simplified0.1

    \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) + 0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)}\right|\]

  15. Final simplification0.1

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) + 0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right|\]

Reproduce

herbie shell --seed 2021084 
(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)))))))