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

Percentage Accurate: 99.8% → 99.9%

Time: 9.7s

Alternatives: 8

Speedup: 4.4×

• 28.4% 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: 99.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r/ 99.3%

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

   2. *-rgt-identity 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-frac-neg 99.3%

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

   6. fabs-div 99.3%

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

   7. associate-/l* 99.3%

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

   8. fabs-div 99.3%

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

7. Simplified 35.2%

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

   1. fma-udef 35.2%

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

9. Applied egg-rr 35.2%

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

    1. fma-udef 35.2%

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

11. Applied egg-rr 35.2%

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

12. Final simplification 35.2%

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

Alternative 2: 99.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.05:\\ \;\;\;\;t\_0 \cdot \left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x_m) 0.05)
     (* t_0 (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0))))
     (* t_0 (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = t_0 * ((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0)));
	} else {
		tmp = t_0 * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x_m) <= 0.05) {
		tmp = t_0 * ((x_m * 2.0) + (0.6666666666666666 * Math.pow(x_m, 3.0)));
	} else {
		tmp = t_0 * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x_m) <= 0.05:
		tmp = t_0 * ((x_m * 2.0) + (0.6666666666666666 * math.pow(x_m, 3.0)))
	else:
		tmp = t_0 * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(t_0 * Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))));
	else
		tmp = Float64(t_0 * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x_m) <= 0.05)
		tmp = t_0 * ((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0)));
	else
		tmp = t_0 * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.05], N[(t$95$0 * N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 99.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. +-commutative 99.0%

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

      4. fabs-neg 99.0%

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

      5. distribute-frac-neg 99.0%

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

      6. fabs-div 99.0%

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

      7. associate-/l* 99.0%

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

      8. fabs-div 99.0%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in x around 0 53.0%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 53.0%

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

      2. associate-*r* 53.0%

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

      3. associate-*r* 53.0%

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

      4. distribute-rgt-out 53.0%

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

   10. Simplified 53.0%

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

   if 0.050000000000000003 < (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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. fabs-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. fabs-div 99.9%

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

      7. associate-/l* 99.9%

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

      8. fabs-div 99.9%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 0.1%

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

   10. Simplified 0.1%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 5.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (/
   (+
    (* 0.047619047619047616 (pow x_m 6.0))
    (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))))
   (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (((0.047619047619047616 * pow(x_m, 6.0)) + (2.0 + (0.6666666666666666 * pow(x_m, 2.0)))) / sqrt(((double) M_PI)));
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (((0.047619047619047616 * math.pow(x_m, 6.0)) + (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0)))) / math.sqrt(math.pi))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(Float64(Float64(0.047619047619047616 * (x_m ^ 6.0)) + Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))) / sqrt(pi)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * (((0.047619047619047616 * (x_m ^ 6.0)) + (2.0 + (0.6666666666666666 * (x_m ^ 2.0)))) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r/ 99.3%

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

   2. *-rgt-identity 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-frac-neg 99.3%

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

   6. fabs-div 99.3%

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

   7. associate-/l* 99.3%

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

   8. fabs-div 99.3%

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

7. Simplified 35.2%

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

   1. fma-udef 35.2%

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

9. Applied egg-rr 35.2%

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

10. Taylor expanded in x around inf 35.0%

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

11. Final simplification 35.0%

    \[\leadsto x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 4: 98.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right)}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (/ 1.0 (/ (sqrt PI) (fma 0.047619047619047616 (pow x_m 6.0) 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (1.0 / (sqrt(((double) M_PI)) / fma(0.047619047619047616, pow(x_m, 6.0), 2.0)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(1.0 / Float64(sqrt(pi) / fma(0.047619047619047616, (x_m ^ 6.0), 2.0))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 98.8%

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

6. Taylor expanded in x around 0 98.5%

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

   1. div-inv 99.1%

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

   2. add-sqr-sqrt 33.4%

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

   3. fabs-sqr 33.4%

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

   4. add-sqr-sqrt 34.8%

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

   5. add-sqr-sqrt 34.8%

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

   6. fabs-sqr 34.8%

      \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}} \cdot \sqrt{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}}} \]

   7. add-sqr-sqrt 34.8%

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. fma-def 34.8%

      \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}} \]

8. Applied egg-rr 34.8%

   \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}} \]

9. Final simplification 34.8%

   \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}} \]
10. Add Preprocessing

Alternative 5: 98.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 98.8%

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

6. Taylor expanded in x around 0 98.5%

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

7. Taylor expanded in x around 0 98.5%

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

   1. associate-*r/ 98.5%

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

   2. +-commutative 98.5%

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

   3. fma-udef 98.5%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right|} \]

   4. fabs-div 98.5%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\frac{\left|\sqrt{\pi} \cdot 1\right|}{\left|\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right|}}} \]

   5. *-rgt-identity 98.5%

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

   6. fabs-div 98.5%

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

   7. fabs-div 98.5%

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

   8. rem-square-sqrt 33.2%

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

   9. fabs-sqr 33.2%

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

   10. rem-square-sqrt 34.6%

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

   11. associate-/r/ 34.6%

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

   12. *-commutative 34.6%

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

9. Simplified 34.6%

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

10. Final simplification 34.6%

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

Alternative 6: 98.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;\left(x\_m \cdot 2\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* (* x_m 2.0) (pow PI -0.5))
   (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = (x_m * 2.0) * pow(((double) M_PI), -0.5);
	} else {
		tmp = 0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x_m, 7.0));
	}
	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 * 2.0) * Math.pow(Math.PI, -0.5);
	} else {
		tmp = 0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = (x_m * 2.0) * math.pow(math.pi, -0.5)
	else:
		tmp = 0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(Float64(x_m * 2.0) * (pi ^ -0.5));
	else
		tmp = Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x_m ^ 7.0)));
	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 * 2.0) * (pi ^ -0.5);
	else
		tmp = 0.047619047619047616 * (sqrt((1.0 / pi)) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(N[(x$95$m * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. fabs-div 99.3%

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

      7. associate-/l* 99.3%

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

      8. fabs-div 99.3%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in x around 0 34.9%

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

      1. associate-*r* 34.9%

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

   10. Simplified 34.9%

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

       1. expm1-log1p-u 34.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

       2. expm1-udef 4.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

       3. associate-*l* 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       4. inv-pow 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1 \]

       5. sqrt-pow1 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1 \]

       6. metadata-eval 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1 \]

   12. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 34.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 34.9%

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

       3. *-commutative 34.9%

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

       4. *-commutative 34.9%

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

       5. associate-*r* 34.9%

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

       6. *-commutative 34.9%

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

   14. Simplified 34.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. fabs-div 99.3%

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

      7. associate-/l* 99.3%

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

      8. fabs-div 99.3%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in x around inf 3.7%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;\left(x\_m \cdot 2\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* (* x_m 2.0) (pow PI -0.5))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = (x_m * 2.0) * pow(((double) M_PI), -0.5);
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	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 * 2.0) * Math.pow(Math.PI, -0.5);
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = (x_m * 2.0) * math.pow(math.pi, -0.5)
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(Float64(x_m * 2.0) * (pi ^ -0.5));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	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 * 2.0) * (pi ^ -0.5);
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(N[(x$95$m * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. fabs-div 99.3%

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

      7. associate-/l* 99.3%

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

      8. fabs-div 99.3%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in x around 0 34.9%

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

      1. associate-*r* 34.9%

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

   10. Simplified 34.9%

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

       1. expm1-log1p-u 34.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

       2. expm1-udef 4.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

       3. associate-*l* 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       4. inv-pow 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1 \]

       5. sqrt-pow1 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1 \]

       6. metadata-eval 4.2%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1 \]

   12. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 34.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 34.9%

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

       3. *-commutative 34.9%

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

       4. *-commutative 34.9%

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

       5. associate-*r* 34.9%

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

       6. *-commutative 34.9%

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

   14. Simplified 34.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. fabs-div 99.3%

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

      7. associate-/l* 99.3%

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

      8. fabs-div 99.3%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in x around inf 3.7%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 3.7%

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

   10. Simplified 3.7%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 17.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m 2.0) (pow PI -0.5)))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * 2.0) * pow(((double) M_PI), -0.5);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * 2.0) * Math.pow(Math.PI, -0.5);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * 2.0) * math.pow(math.pi, -0.5)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * 2.0) * (pi ^ -0.5))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * 2.0) * (pi ^ -0.5);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r/ 99.3%

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

   2. *-rgt-identity 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-frac-neg 99.3%

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

   6. fabs-div 99.3%

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

   7. associate-/l* 99.3%

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

   8. fabs-div 99.3%

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

7. Simplified 35.2%

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

8. Taylor expanded in x around 0 34.9%

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

   1. associate-*r* 34.9%

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

10. Simplified 34.9%

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

    1. expm1-log1p-u 34.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. expm1-udef 4.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

    3. associate-*l* 4.2%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

    4. inv-pow 4.2%

       \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1 \]

    5. sqrt-pow1 4.2%

       \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1 \]

    6. metadata-eval 4.2%

       \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1 \]

12. Applied egg-rr 4.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 34.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\right)} \]

    2. expm1-log1p 34.9%

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

    3. *-commutative 34.9%

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

    4. *-commutative 34.9%

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

    5. associate-*r* 34.9%

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

    6. *-commutative 34.9%

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

14. Simplified 34.9%

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

15. Final simplification 34.9%

    \[\leadsto \left(x \cdot 2\right) \cdot {\pi}^{-0.5} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))))))