Average Error: 0.1 → 0.1

Time: 1.6min

Precision: binary64

Cost: 33472

\[x \leq 0.5\]

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

↓

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

\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|\left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot \frac{1}{\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
  (*
   (*
    (fabs x)
    (+
     (* (pow x 6.0) 0.047619047619047616)
     (+ 2.0 (+ (* 0.6666666666666666 (* x x)) (* 0.2 (pow x 4.0))))))
   (/ 1.0 (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((fabs(x) * ((pow(x, 6.0) * 0.047619047619047616) + (2.0 + ((0.6666666666666666 * (x * x)) + (0.2 * pow(x, 4.0)))))) * (1.0 / sqrt((double) M_PI)));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.1
Cost	33344

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

Alternative 2
Error	0.5
Cost	33344

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

Alternative 3
Error	0.5
Cost	40065

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.025004699991044942:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right|\\ \end{array}\]

Alternative 4
Error	0.7
Cost	33473

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.025004699991044942:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^{7}\right)\right|\\ \end{array}\]

Alternative 5
Error	0.8
Cost	32833

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.025004699991044942:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^{7}\right)\right|\\ \end{array}\]

Alternative 6
Error	4.5
Cost	20032

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\]

Alternative 7
Error	4.8
Cost	19648

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

Alternative 8
Error	4.8
Cost	19648

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

Alternative 9
Error	5.3
Cost	19520

\[\left|\frac{\left|x\right| \cdot 2}{\sqrt{\pi}}\right|\]

Alternative 10
Error	60.5
Cost	64

\[1\]

Derivation

Initial program 0.1
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
Simplified0.1
\[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right|}\]
Using strategy rm
Applied div-inv_binary64_17800.1
\[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right)}\right|\]
Applied associate-*r*_binary64_17230.1
\[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}}\right|\]
Simplified0.1
\[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + {x}^{4} \cdot 0.2\right)\right)\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right|\]
Using strategy rm
Applied *-commutative_binary64_17140.1
\[\leadsto \left|\left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + \color{blue}{0.2 \cdot {x}^{4}}\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
Simplified0.1
\[\leadsto \color{blue}{\left|\left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|}\]
Final simplification0.1
\[\leadsto \left|\left(\left|x\right| \cdot \left({x}^{6} \cdot 0.047619047619047616 + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

Reproduce

herbie shell --seed 2021065 
(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)))))))