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

Percentage Accurate: 99.8% → 99.8%

Time: 11.2s

Alternatives: 8

Speedup: 4.4×

• Could not converge on a ground truth

  1. reg0 = (bf 808362734901008000)

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, 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] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, 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] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot \left(0.047619047619047616 \cdot {x\_m}^{6} + \mathsf{fma}\left(0.6666666666666666, {x\_m}^{2}, 2\right)\right) + x\_m \cdot \left(0.2 \cdot {x\_m}^{4}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (*
    x_m
    (+
     (* 0.047619047619047616 (pow x_m 6.0))
     (fma 0.6666666666666666 (pow x_m 2.0) 2.0)))
   (* x_m (* 0.2 (pow x_m 4.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * ((x_m * ((0.047619047619047616 * pow(x_m, 6.0)) + fma(0.6666666666666666, pow(x_m, 2.0), 2.0))) + (x_m * (0.2 * pow(x_m, 4.0))));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(x_m * Float64(Float64(0.047619047619047616 * (x_m ^ 6.0)) + fma(0.6666666666666666, (x_m ^ 2.0), 2.0))) + Float64(x_m * Float64(0.2 * (x_m ^ 4.0)))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(x$95$m * N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow1 99.8%

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

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 38.4%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

   2. associate-*r* 38.4%

      \[\leadsto \color{blue}{\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)} \]

   3. *-commutative 38.4%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right)} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   4. associate-*l* 38.4%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

   5. fma-undefine 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \color{blue}{\left({x}^{6} \cdot 0.047619047619047616 + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right)\right) \]

   6. *-commutative 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right) \]

   7. associate-+r+ 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right) \]

   8. +-commutative 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   9. fma-define 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   10. fma-undefine 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + {x}^{4} \cdot 0.2\right)}\right)\right) \]

   11. *-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + {x}^{4} \cdot 0.2\right)\right)\right) \]

   12. fma-define 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, {x}^{4} \cdot 0.2\right)}\right)\right) \]

   13. *-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, \color{blue}{0.2 \cdot {x}^{4}}\right)\right)\right) \]

9. Simplified 38.4%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, 0.2 \cdot {x}^{4}\right)\right)\right)} \]
10. Step-by-step derivation

    1. fma-undefine 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)}\right)\right) \]

11. Applied egg-rr 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)}\right)\right) \]
12. Step-by-step derivation

    1. associate-+r+ 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + {x}^{2} \cdot 0.6666666666666666\right) + 0.2 \cdot {x}^{4}\right)}\right) \]

    2. distribute-rgt-in 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + {x}^{2} \cdot 0.6666666666666666\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)} \]

    3. fma-undefine 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + {x}^{2} \cdot 0.6666666666666666\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right) \]

    4. *-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\left(\left(0.047619047619047616 \cdot {x}^{6} + 2\right) + \color{blue}{0.6666666666666666 \cdot {x}^{2}}\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right) \]

    5. associate-+l+ 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right) \]

    6. +-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right) \]

    7. fma-define 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right) \]

13. Applied egg-rr 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)} \]

14. Final simplification 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) + x \cdot \left(0.2 \cdot {x}^{4}\right)\right) \]
15. Add Preprocessing

Alternative 2: 99.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot \left(\left(0.047619047619047616 \cdot {x\_m}^{6} + 2\right) + \left(0.2 \cdot {x\_m}^{4} + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x_m
   (+
    (+ (* 0.047619047619047616 (pow x_m 6.0)) 2.0)
    (+ (* 0.2 (pow x_m 4.0)) (* 0.6666666666666666 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * (((0.047619047619047616 * pow(x_m, 6.0)) + 2.0) + ((0.2 * pow(x_m, 4.0)) + (0.6666666666666666 * pow(x_m, 2.0)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * (((0.047619047619047616 * Math.pow(x_m, 6.0)) + 2.0) + ((0.2 * Math.pow(x_m, 4.0)) + (0.6666666666666666 * Math.pow(x_m, 2.0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * (((0.047619047619047616 * math.pow(x_m, 6.0)) + 2.0) + ((0.2 * math.pow(x_m, 4.0)) + (0.6666666666666666 * math.pow(x_m, 2.0)))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * Float64(Float64(Float64(0.047619047619047616 * (x_m ^ 6.0)) + 2.0) + Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(0.6666666666666666 * (x_m ^ 2.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * (((0.047619047619047616 * (x_m ^ 6.0)) + 2.0) + ((0.2 * (x_m ^ 4.0)) + (0.6666666666666666 * (x_m ^ 2.0)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow1 99.8%

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

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 38.4%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

   2. associate-*r* 38.4%

      \[\leadsto \color{blue}{\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)} \]

   3. *-commutative 38.4%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right)} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   4. associate-*l* 38.4%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

   5. fma-undefine 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \color{blue}{\left({x}^{6} \cdot 0.047619047619047616 + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right)\right) \]

   6. *-commutative 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right) \]

   7. associate-+r+ 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right) \]

   8. +-commutative 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   9. fma-define 38.4%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

   10. fma-undefine 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + {x}^{4} \cdot 0.2\right)}\right)\right) \]

   11. *-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + {x}^{4} \cdot 0.2\right)\right)\right) \]

   12. fma-define 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, {x}^{4} \cdot 0.2\right)}\right)\right) \]

   13. *-commutative 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, \color{blue}{0.2 \cdot {x}^{4}}\right)\right)\right) \]

9. Simplified 38.4%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, 0.2 \cdot {x}^{4}\right)\right)\right)} \]
10. Step-by-step derivation

    1. fma-undefine 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)}\right)\right) \]

11. Applied egg-rr 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)}\right)\right) \]
12. Step-by-step derivation

    1. fma-define 38.4%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + \left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)\right)\right) \]

13. Applied egg-rr 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + \left({x}^{2} \cdot 0.6666666666666666 + 0.2 \cdot {x}^{4}\right)\right)\right) \]

14. Final simplification 38.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 98.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m\right| \cdot \left|\frac{0.047619047619047616 \cdot {x\_m}^{6} + 2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fabs x_m)
  (fabs (/ (+ (* 0.047619047619047616 (pow x_m 6.0)) 2.0) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(x_m) * fabs((((0.047619047619047616 * pow(x_m, 6.0)) + 2.0) / sqrt(((double) M_PI))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.abs(x_m) * Math.abs((((0.047619047619047616 * Math.pow(x_m, 6.0)) + 2.0) / Math.sqrt(Math.PI)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(x_m) * math.fabs((((0.047619047619047616 * math.pow(x_m, 6.0)) + 2.0) / math.sqrt(math.pi)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(abs(x_m) * abs(Float64(Float64(Float64(0.047619047619047616 * (x_m ^ 6.0)) + 2.0) / sqrt(pi))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(x_m) * abs((((0.047619047619047616 * (x_m ^ 6.0)) + 2.0) / sqrt(pi)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 98.6%

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

6. Taylor expanded in x around inf 98.5%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
7. Add Preprocessing

Alternative 4: 99.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.5:\\ \;\;\;\;x\_m \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \sqrt{\frac{0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (* x_m (* (pow PI -0.5) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (* (pow x_m 7.0) (sqrt (/ 0.0022675736961451248 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = x_m * (pow(((double) M_PI), -0.5) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = pow(x_m, 7.0) * sqrt((0.0022675736961451248 / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = x_m * (Math.pow(Math.PI, -0.5) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.pow(x_m, 7.0) * Math.sqrt((0.0022675736961451248 / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = x_m * (math.pow(math.pi, -0.5) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.pow(x_m, 7.0) * math.sqrt((0.0022675736961451248 / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(x_m * Float64((pi ^ -0.5) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64((x_m ^ 7.0) * sqrt(Float64(0.0022675736961451248 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = x_m * ((pi ^ -0.5) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = (x_m ^ 7.0) * sqrt((0.0022675736961451248 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[Sqrt[N[(0.0022675736961451248 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. pow1 99.8%

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

   7. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 54.9%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

      2. associate-*r* 54.8%

         \[\leadsto \color{blue}{\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)} \]

      3. *-commutative 54.8%

         \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right)} \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

      4. associate-*l* 54.8%

         \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right)} \]

      5. fma-undefine 54.8%

         \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \color{blue}{\left({x}^{6} \cdot 0.047619047619047616 + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right)\right) \]

      6. *-commutative 54.8%

         \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right)\right) \]

      7. associate-+r+ 54.8%

         \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)}\right) \]

      8. +-commutative 54.8%

         \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

      9. fma-define 54.8%

         \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, {x}^{4} \cdot 0.2\right)\right)\right) \]

      10. fma-undefine 54.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + {x}^{4} \cdot 0.2\right)}\right)\right) \]

      11. *-commutative 54.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + {x}^{4} \cdot 0.2\right)\right)\right) \]

      12. fma-define 54.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, {x}^{4} \cdot 0.2\right)}\right)\right) \]

      13. *-commutative 54.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, \color{blue}{0.2 \cdot {x}^{4}}\right)\right)\right) \]

   9. Simplified 54.8%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) + \mathsf{fma}\left({x}^{2}, 0.6666666666666666, 0.2 \cdot {x}^{4}\right)\right)\right)} \]

   10. Taylor expanded in x around 0 54.5%

       \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 54.5%

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

       2. associate-*r* 54.5%

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

       3. *-commutative 54.5%

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

       4. distribute-rgt-out 54.5%

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

       5. *-commutative 54.5%

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

   12. Simplified 54.5%

       \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 54.5%

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

       2. inv-pow 54.5%

          \[\leadsto x \cdot \left(\left(1 \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

       3. sqrt-pow1 54.5%

          \[\leadsto x \cdot \left(\left(1 \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right) \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

       4. metadata-eval 54.5%

          \[\leadsto x \cdot \left(\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

   14. Applied egg-rr 54.5%

       \[\leadsto x \cdot \left(\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]
   15. Step-by-step derivation

       1. *-lft-identity 54.5%

          \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

   16. Simplified 54.5%

       \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 99.0%

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

      2. associate-*l* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*l* 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left|0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
   9. Step-by-step derivation

      1. *-commutative 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow-sqr 98.9%

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

      4. fabs-sqr 98.9%

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

      5. pow-sqr 99.0%

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

      6. metadata-eval 99.0%

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

      7. fabs-mul 99.0%

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

      8. unpow-1 99.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot 0.047619047619047616\right| \]

      9. metadata-eval 99.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot 0.047619047619047616\right| \]

      10. pow-sqr 99.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right) \cdot 0.047619047619047616\right| \]

      11. rem-sqrt-square 99.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right) \cdot 0.047619047619047616\right| \]

      12. rem-square-sqrt 99.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right) \cdot 0.047619047619047616\right| \]

      13. fabs-sqr 99.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right) \cdot 0.047619047619047616\right| \]

      14. rem-square-sqrt 99.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{{\pi}^{-0.5}}\right) \cdot 0.047619047619047616\right| \]

      15. associate-*r* 99.0%

          \[\leadsto \left|\color{blue}{\left|{x}^{6} \cdot x\right| \cdot \left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right| \]

   10. Simplified 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.1%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{0.047619047619047616 \cdot {\pi}^{-0.5}} \cdot \sqrt{0.047619047619047616 \cdot {\pi}^{-0.5}}\right)} \]

       2. sqrt-unprod 0.1%

          \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)}} \]

       3. *-commutative 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}} \]

       4. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right)} \]

       5. sqrt-pow1 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right)} \]

       6. inv-pow 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right)} \]

       7. *-commutative 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       8. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       9. sqrt-pow1 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       10. inv-pow 0.1%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       11. swap-sqr 0.1%

           \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}} \]

       12. add-sqr-sqrt 0.1%

           \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)} \]

       13. metadata-eval 0.1%

           \[\leadsto {x}^{7} \cdot \sqrt{\frac{1}{\pi} \cdot \color{blue}{0.0022675736961451248}} \]

   12. Applied egg-rr 0.1%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot 0.0022675736961451248}} \]
   13. Step-by-step derivation

       1. associate-*l/ 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1 \cdot 0.0022675736961451248}{\pi}}} \]

       2. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \sqrt{\frac{\color{blue}{0.0022675736961451248}}{\pi}} \]

   14. Simplified 0.1%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\frac{0.0022675736961451248}{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x\_m \cdot \mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right)}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (* x_m (fma 0.047619047619047616 (pow x_m 6.0) 2.0)) (sqrt PI)))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * fma(0.047619047619047616, pow(x_m, 6.0), 2.0)) / sqrt(((double) M_PI));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * fma(0.047619047619047616, (x_m ^ 6.0), 2.0)) / sqrt(pi))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 98.6%

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

6. Taylor expanded in x around inf 98.5%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 36.4%

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

   2. fabs-sqr 36.4%

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

   3. add-sqr-sqrt 38.0%

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

   4. add-sqr-sqrt 37.4%

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

   5. fabs-sqr 37.4%

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

   6. add-sqr-sqrt 38.0%

      \[\leadsto x \cdot \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}} \]

   7. associate-*r/ 37.8%

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

   8. fma-define 37.8%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}{\sqrt{\pi}} \]

8. Applied egg-rr 37.8%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}{\sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 6: 98.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \sqrt{\frac{0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (sqrt (/ 4.0 PI)))
   (* (pow x_m 7.0) (sqrt (/ 0.0022675736961451248 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * sqrt((0.0022675736961451248 / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * Math.sqrt((0.0022675736961451248 / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * math.sqrt((4.0 / math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * math.sqrt((0.0022675736961451248 / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * sqrt(Float64(4.0 / pi)));
	else
		tmp = Float64((x_m ^ 7.0) * sqrt(Float64(0.0022675736961451248 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * sqrt((4.0 / pi));
	else
		tmp = (x_m ^ 7.0) * sqrt((0.0022675736961451248 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[Sqrt[N[(0.0022675736961451248 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.5%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-*l* 70.5%

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

   7. Simplified 70.5%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 70.1%

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

      2. fabs-sqr 70.1%

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

      3. add-sqr-sqrt 70.5%

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

      4. *-commutative 70.5%

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

      5. inv-pow 70.5%

         \[\leadsto \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right) \cdot \left|x\right| \]

      6. sqrt-pow1 70.5%

         \[\leadsto \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot 2\right) \cdot \left|x\right| \]

      7. metadata-eval 70.5%

         \[\leadsto \left({\pi}^{\color{blue}{-0.5}} \cdot 2\right) \cdot \left|x\right| \]

      8. add-sqr-sqrt 36.4%

         \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \]

      9. fabs-sqr 36.4%

         \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      10. add-sqr-sqrt 38.1%

          \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{x} \]

   9. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot 2\right) \cdot x} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 37.5%

          \[\leadsto \color{blue}{\left(\sqrt{{\pi}^{-0.5} \cdot 2} \cdot \sqrt{{\pi}^{-0.5} \cdot 2}\right)} \cdot x \]

       2. sqrt-unprod 38.1%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot 2\right)}} \cdot x \]

       3. *-commutative 38.1%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \cdot \left({\pi}^{-0.5} \cdot 2\right)} \cdot x \]

       4. *-commutative 38.1%

          \[\leadsto \sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)}} \cdot x \]

       5. swap-sqr 38.1%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \cdot x \]

       6. metadata-eval 38.1%

          \[\leadsto \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot x \]

       7. pow-prod-up 38.1%

          \[\leadsto \sqrt{4 \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}} \cdot x \]

       8. metadata-eval 38.1%

          \[\leadsto \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \cdot x \]

   11. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{\sqrt{4 \cdot {\pi}^{-1}}} \cdot x \]
   12. Step-by-step derivation

       1. unpow-1 38.1%

          \[\leadsto \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \cdot x \]

       2. associate-*r/ 38.1%

          \[\leadsto \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \cdot x \]

       3. metadata-eval 38.1%

          \[\leadsto \sqrt{\frac{\color{blue}{4}}{\pi}} \cdot x \]

   13. Simplified 38.1%

       \[\leadsto \color{blue}{\sqrt{\frac{4}{\pi}}} \cdot x \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 34.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 34.0%

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

      2. associate-*l* 33.9%

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

      3. *-commutative 33.9%

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

      4. associate-*r* 34.0%

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

      5. *-commutative 34.0%

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

      6. associate-*l* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in x around 0 34.0%

      \[\leadsto \color{blue}{\left|0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
   9. Step-by-step derivation

      1. *-commutative 34.0%

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

      2. metadata-eval 34.0%

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

      3. pow-sqr 33.9%

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

      4. fabs-sqr 33.9%

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

      5. pow-sqr 34.0%

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

      6. metadata-eval 34.0%

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

      7. fabs-mul 34.0%

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

      8. unpow-1 34.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot 0.047619047619047616\right| \]

      9. metadata-eval 34.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot 0.047619047619047616\right| \]

      10. pow-sqr 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right) \cdot 0.047619047619047616\right| \]

      11. rem-sqrt-square 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right) \cdot 0.047619047619047616\right| \]

      12. rem-square-sqrt 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right) \cdot 0.047619047619047616\right| \]

      13. fabs-sqr 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right) \cdot 0.047619047619047616\right| \]

      14. rem-square-sqrt 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{{\pi}^{-0.5}}\right) \cdot 0.047619047619047616\right| \]

      15. associate-*r* 34.0%

          \[\leadsto \left|\color{blue}{\left|{x}^{6} \cdot x\right| \cdot \left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right| \]

   10. Simplified 4.0%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 4.0%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{0.047619047619047616 \cdot {\pi}^{-0.5}} \cdot \sqrt{0.047619047619047616 \cdot {\pi}^{-0.5}}\right)} \]

       2. sqrt-unprod 4.0%

          \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)}} \]

       3. *-commutative 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}} \]

       4. metadata-eval 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right)} \]

       5. sqrt-pow1 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right)} \]

       6. inv-pow 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right)} \]

       7. *-commutative 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       8. metadata-eval 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       9. sqrt-pow1 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       10. inv-pow 4.0%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

       11. swap-sqr 4.0%

           \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}} \]

       12. add-sqr-sqrt 4.0%

           \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)} \]

       13. metadata-eval 4.0%

           \[\leadsto {x}^{7} \cdot \sqrt{\frac{1}{\pi} \cdot \color{blue}{0.0022675736961451248}} \]

   12. Applied egg-rr 4.0%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot 0.0022675736961451248}} \]
   13. Step-by-step derivation

       1. associate-*l/ 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1 \cdot 0.0022675736961451248}{\pi}}} \]

       2. metadata-eval 4.0%

          \[\leadsto {x}^{7} \cdot \sqrt{\frac{\color{blue}{0.0022675736961451248}}{\pi}} \]

   14. Simplified 4.0%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\frac{0.0022675736961451248}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \sqrt{\frac{0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x\_m}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (sqrt (/ 4.0 PI)))
   (sqrt (* 0.0022675736961451248 (/ (pow x_m 14.0) PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = sqrt((0.0022675736961451248 * (pow(x_m, 14.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = Math.sqrt((0.0022675736961451248 * (Math.pow(x_m, 14.0) / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * math.sqrt((4.0 / math.pi))
	else:
		tmp = math.sqrt((0.0022675736961451248 * (math.pow(x_m, 14.0) / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * sqrt(Float64(4.0 / pi)));
	else
		tmp = sqrt(Float64(0.0022675736961451248 * Float64((x_m ^ 14.0) / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * sqrt((4.0 / pi));
	else
		tmp = sqrt((0.0022675736961451248 * ((x_m ^ 14.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x$95$m, 14.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.5%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-*l* 70.5%

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

   7. Simplified 70.5%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 70.1%

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

      2. fabs-sqr 70.1%

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

      3. add-sqr-sqrt 70.5%

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

      4. *-commutative 70.5%

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

      5. inv-pow 70.5%

         \[\leadsto \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right) \cdot \left|x\right| \]

      6. sqrt-pow1 70.5%

         \[\leadsto \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot 2\right) \cdot \left|x\right| \]

      7. metadata-eval 70.5%

         \[\leadsto \left({\pi}^{\color{blue}{-0.5}} \cdot 2\right) \cdot \left|x\right| \]

      8. add-sqr-sqrt 36.4%

         \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \]

      9. fabs-sqr 36.4%

         \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      10. add-sqr-sqrt 38.1%

          \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{x} \]

   9. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot 2\right) \cdot x} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 37.5%

          \[\leadsto \color{blue}{\left(\sqrt{{\pi}^{-0.5} \cdot 2} \cdot \sqrt{{\pi}^{-0.5} \cdot 2}\right)} \cdot x \]

       2. sqrt-unprod 38.1%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot 2\right)}} \cdot x \]

       3. *-commutative 38.1%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \cdot \left({\pi}^{-0.5} \cdot 2\right)} \cdot x \]

       4. *-commutative 38.1%

          \[\leadsto \sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)}} \cdot x \]

       5. swap-sqr 38.1%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \cdot x \]

       6. metadata-eval 38.1%

          \[\leadsto \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot x \]

       7. pow-prod-up 38.1%

          \[\leadsto \sqrt{4 \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}} \cdot x \]

       8. metadata-eval 38.1%

          \[\leadsto \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \cdot x \]

   11. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{\sqrt{4 \cdot {\pi}^{-1}}} \cdot x \]
   12. Step-by-step derivation

       1. unpow-1 38.1%

          \[\leadsto \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \cdot x \]

       2. associate-*r/ 38.1%

          \[\leadsto \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \cdot x \]

       3. metadata-eval 38.1%

          \[\leadsto \sqrt{\frac{\color{blue}{4}}{\pi}} \cdot x \]

   13. Simplified 38.1%

       \[\leadsto \color{blue}{\sqrt{\frac{4}{\pi}}} \cdot x \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 34.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 34.0%

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

      2. associate-*l* 33.9%

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

      3. *-commutative 33.9%

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

      4. associate-*r* 34.0%

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

      5. *-commutative 34.0%

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

      6. associate-*l* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in x around 0 34.0%

      \[\leadsto \color{blue}{\left|0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
   9. Step-by-step derivation

      1. *-commutative 34.0%

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

      2. metadata-eval 34.0%

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

      3. pow-sqr 33.9%

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

      4. fabs-sqr 33.9%

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

      5. pow-sqr 34.0%

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

      6. metadata-eval 34.0%

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

      7. fabs-mul 34.0%

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

      8. unpow-1 34.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot 0.047619047619047616\right| \]

      9. metadata-eval 34.0%

         \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot 0.047619047619047616\right| \]

      10. pow-sqr 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right) \cdot 0.047619047619047616\right| \]

      11. rem-sqrt-square 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right) \cdot 0.047619047619047616\right| \]

      12. rem-square-sqrt 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right) \cdot 0.047619047619047616\right| \]

      13. fabs-sqr 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right) \cdot 0.047619047619047616\right| \]

      14. rem-square-sqrt 34.0%

          \[\leadsto \left|\left(\left|{x}^{6} \cdot x\right| \cdot \color{blue}{{\pi}^{-0.5}}\right) \cdot 0.047619047619047616\right| \]

      15. associate-*r* 34.0%

          \[\leadsto \left|\color{blue}{\left|{x}^{6} \cdot x\right| \cdot \left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right| \]

   10. Simplified 4.0%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 3.9%

          \[\leadsto \color{blue}{\sqrt{{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)}} \]

       2. sqrt-unprod 31.4%

          \[\leadsto \color{blue}{\sqrt{\left({x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \cdot \left({x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)}} \]

       3. swap-sqr 31.4%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{7} \cdot {x}^{7}\right) \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)}} \]

       4. pow-prod-up 31.4%

          \[\leadsto \sqrt{\color{blue}{{x}^{\left(7 + 7\right)}} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       5. metadata-eval 31.4%

          \[\leadsto \sqrt{{x}^{\color{blue}{14}} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       6. *-commutative 31.4%

          \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right)} \]

       7. metadata-eval 31.4%

          \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right)\right)} \]

       8. sqrt-pow1 31.4%

          \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right)\right)} \]

       9. inv-pow 31.4%

          \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right)\right)} \]

       10. *-commutative 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)\right)} \]

       11. metadata-eval 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)\right)} \]

       12. sqrt-pow1 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)\right)} \]

       13. inv-pow 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)\right)} \]

       14. swap-sqr 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{\pi}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)}} \]

       15. add-sqr-sqrt 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\color{blue}{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)} \]

       16. metadata-eval 31.4%

           \[\leadsto \sqrt{{x}^{14} \cdot \left(\frac{1}{\pi} \cdot \color{blue}{0.0022675736961451248}\right)} \]

   12. Applied egg-rr 31.4%

       \[\leadsto \color{blue}{\sqrt{{x}^{14} \cdot \left(\frac{1}{\pi} \cdot 0.0022675736961451248\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* 31.4%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{14} \cdot \frac{1}{\pi}\right) \cdot 0.0022675736961451248}} \]

       2. *-commutative 31.4%

          \[\leadsto \sqrt{\color{blue}{0.0022675736961451248 \cdot \left({x}^{14} \cdot \frac{1}{\pi}\right)}} \]

       3. metadata-eval 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left({x}^{\color{blue}{\left(2 \cdot 7\right)}} \cdot \frac{1}{\pi}\right)} \]

       4. pow-sqr 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\color{blue}{\left({x}^{7} \cdot {x}^{7}\right)} \cdot \frac{1}{\pi}\right)} \]

       5. associate-*r/ 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \color{blue}{\frac{\left({x}^{7} \cdot {x}^{7}\right) \cdot 1}{\pi}}} \]

       6. associate-*r* 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{7} \cdot \left({x}^{7} \cdot 1\right)}}{\pi}} \]

       7. *-commutative 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{{x}^{7} \cdot \color{blue}{\left(1 \cdot {x}^{7}\right)}}{\pi}} \]

       8. *-lft-identity 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{{x}^{7} \cdot \color{blue}{{x}^{7}}}{\pi}} \]

       9. pow-sqr 31.4%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi}} \]

       10. metadata-eval 31.4%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{{x}^{\color{blue}{14}}}{\pi}} \]

   14. Simplified 31.4%

       \[\leadsto \color{blue}{\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \sqrt{\frac{4}{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (sqrt (/ 4.0 PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * sqrt((4.0 / ((double) M_PI)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.sqrt((4.0 / Math.PI));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.sqrt((4.0 / math.pi))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * sqrt(Float64(4.0 / pi)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * sqrt((4.0 / pi));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 70.5%

   \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
6. Step-by-step derivation

   1. *-commutative 70.5%

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

   2. *-commutative 70.5%

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

   3. associate-*l* 70.5%

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

7. Simplified 70.5%

   \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
8. Step-by-step derivation

   1. add-sqr-sqrt 70.1%

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

   2. fabs-sqr 70.1%

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

   3. add-sqr-sqrt 70.5%

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

   4. *-commutative 70.5%

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

   5. inv-pow 70.5%

      \[\leadsto \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right) \cdot \left|x\right| \]

   6. sqrt-pow1 70.5%

      \[\leadsto \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot 2\right) \cdot \left|x\right| \]

   7. metadata-eval 70.5%

      \[\leadsto \left({\pi}^{\color{blue}{-0.5}} \cdot 2\right) \cdot \left|x\right| \]

   8. add-sqr-sqrt 36.4%

      \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \]

   9. fabs-sqr 36.4%

      \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   10. add-sqr-sqrt 38.1%

       \[\leadsto \left({\pi}^{-0.5} \cdot 2\right) \cdot \color{blue}{x} \]

9. Applied egg-rr 38.1%

   \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot 2\right) \cdot x} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 37.5%

       \[\leadsto \color{blue}{\left(\sqrt{{\pi}^{-0.5} \cdot 2} \cdot \sqrt{{\pi}^{-0.5} \cdot 2}\right)} \cdot x \]

    2. sqrt-unprod 38.1%

       \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot 2\right)}} \cdot x \]

    3. *-commutative 38.1%

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \cdot \left({\pi}^{-0.5} \cdot 2\right)} \cdot x \]

    4. *-commutative 38.1%

       \[\leadsto \sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)}} \cdot x \]

    5. swap-sqr 38.1%

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \cdot x \]

    6. metadata-eval 38.1%

       \[\leadsto \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot x \]

    7. pow-prod-up 38.1%

       \[\leadsto \sqrt{4 \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}} \cdot x \]

    8. metadata-eval 38.1%

       \[\leadsto \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \cdot x \]

11. Applied egg-rr 38.1%

    \[\leadsto \color{blue}{\sqrt{4 \cdot {\pi}^{-1}}} \cdot x \]
12. Step-by-step derivation

    1. unpow-1 38.1%

       \[\leadsto \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \cdot x \]

    2. associate-*r/ 38.1%

       \[\leadsto \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \cdot x \]

    3. metadata-eval 38.1%

       \[\leadsto \sqrt{\frac{\color{blue}{4}}{\pi}} \cdot x \]

13. Simplified 38.1%

    \[\leadsto \color{blue}{\sqrt{\frac{4}{\pi}}} \cdot x \]

14. Final simplification 38.1%

    \[\leadsto x \cdot \sqrt{\frac{4}{\pi}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))))