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

Average Error: 0.2 → 0.2

Time: 9.6s

Precision: binary64

Cost: 39680

\[x \leq 0.5\]

Math

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

↓

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

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
  (*
   (sqrt (/ 1.0 PI))
   (+
    (* (pow x 3.0) (+ 0.6666666666666666 (* x (* x 0.2))))
    (fma 2.0 x (* 0.047619047619047616 (pow x 7.0)))))))

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((sqrt((1.0 / ((double) M_PI))) * ((pow(x, 3.0) * (0.6666666666666666 + (x * (x * 0.2)))) + fma(2.0, x, (0.047619047619047616 * pow(x, 7.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * Float64(Float64(abs(x) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 5.0) * Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 21.0) * Float64(Float64(Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x))))))
end

↓

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64((x ^ 3.0) * Float64(0.6666666666666666 + Float64(x * Float64(x * 0.2)))) + fma(2.0, x, Float64(0.047619047619047616 * (x ^ 7.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(0.6666666666666666 + N[(x * N[(x * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

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. Simplified 0.6

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

   [Start] 0.2

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

   associate-*l/ [=>] 0.6

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

3. Taylor expanded in x around 0 0.2

   \[\leadsto \left|\color{blue}{2 \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.047619047619047616 \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

4. Simplified 0.2

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

   [Start] 0.2	\[ \left|2 \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.047619047619047616 \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   associate-+r+ [=>] 0.2	\[ \left|\color{blue}{\left(2 \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]
   associate-*r* [=>] 0.2	\[ \left|\left(\color{blue}{\left(2 \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]
   associate-*r* [=>] 0.2	\[ \left|\left(\left(2 \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right) + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]
   distribute-rgt-out [=>] 0.2	\[ \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right)\right)} + \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]
   *-commutative [=>] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right)\right) + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.6666666666666666 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot {x}^{2}\right)}\right| \]

5. Applied egg-rr 0.2

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

6. Simplified 0.2

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

   [Start] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.2 \cdot {x}^{3}\right) + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   associate-*r* [=>] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot {x}^{3}} + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   associate-*r* [=>] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot {x}^{3} + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   unpow3 [<=] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot {x}^{3} + \color{blue}{{x}^{3}} \cdot 0.6666666666666666\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   *-commutative [<=] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot {x}^{3} + \color{blue}{0.6666666666666666 \cdot {x}^{3}}\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   distribute-rgt-out [=>] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.2 + 0.6666666666666666\right)} + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   +-commutative [<=] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left({x}^{3} \cdot \color{blue}{\left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)} + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
   associate-*l* [=>] 0.2	\[ \left|\sqrt{\frac{1}{\pi}} \cdot \left({x}^{3} \cdot \left(0.6666666666666666 + \color{blue}{x \cdot \left(x \cdot 0.2\right)}\right) + \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	33280

\[\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left({x}^{4} \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right)\right)\right| \]

Alternative 2
Error	0.4
Cost	33028

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32900

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|t_0 \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32772

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \end{array} \]

Alternative 5
Error	0.7
Cost	26372

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \end{array} \]

Alternative 6
Error	1.0
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7
Error	5.0
Cost	19456

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

Reproduce

herbie shell --seed 2023046 
(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)))))))