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

Percentage Accurate: 99.8% → 99.8%

Time: 14.7s

Alternatives: 14

Speedup: 4.4×

• 28.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.001:\\ \;\;\;\;x\_m \cdot \frac{2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x\_m}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right) + \left(\frac{2}{{x\_m}^{6}} + \frac{0.6666666666666666}{{x\_m}^{4}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.001)
   (*
    x_m
    (/
     (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (pow x_m 2.0)))))
     (sqrt PI)))
   (*
    (pow x_m 7.0)
    (*
     (sqrt (/ 1.0 PI))
     (+
      (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))
      (+ (/ 2.0 (pow x_m 6.0)) (/ 0.6666666666666666 (pow x_m 4.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.001) {
		tmp = x_m * ((2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * pow(x_m, 2.0))))) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (sqrt((1.0 / ((double) M_PI))) * ((0.047619047619047616 + (0.2 / pow(x_m, 2.0))) + ((2.0 / pow(x_m, 6.0)) + (0.6666666666666666 / pow(x_m, 4.0)))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.001) {
		tmp = x_m * ((2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x_m, 2.0))))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.sqrt((1.0 / Math.PI)) * ((0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))) + ((2.0 / Math.pow(x_m, 6.0)) + (0.6666666666666666 / Math.pow(x_m, 4.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.001:
		tmp = x_m * ((2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x_m, 2.0))))) / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (math.sqrt((1.0 / math.pi)) * ((0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))) + ((2.0 / math.pow(x_m, 6.0)) + (0.6666666666666666 / math.pow(x_m, 4.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.001)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x_m ^ 2.0))))) / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0))) + Float64(Float64(2.0 / (x_m ^ 6.0)) + Float64(0.6666666666666666 / (x_m ^ 4.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.001)
		tmp = x_m * ((2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m ^ 2.0))))) / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (sqrt((1.0 / pi)) * ((0.047619047619047616 + (0.2 / (x_m ^ 2.0))) + ((2.0 / (x_m ^ 6.0)) + (0.6666666666666666 / (x_m ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.001], N[(x$95$m * N[(N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-3

   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 99.9%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. *-commutative 93.1%

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

   7. Simplified 93.1%

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

      1. expm1-log1p-u 93.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)\right)} \]

      2. expm1-undefine 34.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)} - 1} \]

   9. Applied egg-rr 4.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 4.0%

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

       2. rem-exp-log 4.0%

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

       3. +-commutative 4.0%

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

       4. associate--l+ 29.8%

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

       5. metadata-eval 29.8%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

       6. metadata-eval 29.8%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

       7. sub-neg 29.8%

          \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

       8. --rgt-identity 29.8%

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

   11. Simplified 29.8%

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

   12. Taylor expanded in x around 0 29.8%

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

   if 1e-3 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 98.4%

         \[\leadsto {x}^{7} \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

      2. associate-*r* 98.4%

         \[\leadsto {x}^{7} \cdot \left(\left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) + \left(0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      3. distribute-rgt-out 98.4%

         \[\leadsto {x}^{7} \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)} + \left(0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      4. associate-*r* 98.4%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right) + \left(\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      5. associate-*r* 98.4%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right) + \left(\left(0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \frac{1}{{x}^{4}} + 2 \cdot \frac{1}{{x}^{6}}\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right) + \left(\frac{2}{{x}^{6}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.001:\\ \;\;\;\;x \cdot \frac{2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right) + \left(\frac{2}{{x}^{6}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x\_m}^{6} + \left(0.2 \cdot {x\_m}^{4} + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fabs x_m)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x_m 6.0))
      (+ (* 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 fabs(x_m) * fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 + ((0.047619047619047616 * pow(x_m, 6.0)) + ((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.abs(x_m) * Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 + ((0.047619047619047616 * Math.pow(x_m, 6.0)) + ((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.fabs(x_m) * math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 + ((0.047619047619047616 * math.pow(x_m, 6.0)) + ((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(abs(x_m) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x_m ^ 6.0)) + 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 = abs(x_m) * abs((sqrt((1.0 / pi)) * (2.0 + ((0.047619047619047616 * (x_m ^ 6.0)) + ((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[Abs[x$95$m], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $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]], $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 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $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 99.4%

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

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 28.7%

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

   3. fabs-sqr 28.7%

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

   4. add-sqr-sqrt 99.8%

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

   5. pow1/2 99.8%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. add-sqr-sqrt 28.7%

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

   2. fabs-sqr 28.7%

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

   3. add-sqr-sqrt 28.3%

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

   4. fabs-sqr 28.3%

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

   5. add-sqr-sqrt 29.8%

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

   6. add-sqr-sqrt 30.3%

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

   7. clear-num 30.3%

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

6. Applied egg-rr 30.1%

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

7. Final simplification 30.1%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

6. Applied egg-rr 30.1%

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

7. Final simplification 30.1%

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

Alternative 6: 99.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.002:\\ \;\;\;\;x\_m \cdot \frac{2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x\_m}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right) + {x\_m}^{6} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.002)
   (*
    x_m
    (/
     (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (pow x_m 2.0)))))
     (sqrt PI)))
   (/
    (*
     x_m
     (+
      (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))
      (* (pow x_m 6.0) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0))))))
    (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = x_m * ((2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * pow(x_m, 2.0))))) / sqrt(((double) M_PI)));
	} else {
		tmp = (x_m * ((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) + (pow(x_m, 6.0) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0)))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = x_m * ((2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x_m, 2.0))))) / Math.sqrt(Math.PI));
	} else {
		tmp = (x_m * ((2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))) + (Math.pow(x_m, 6.0) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0)))))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.002:
		tmp = x_m * ((2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x_m, 2.0))))) / math.sqrt(math.pi))
	else:
		tmp = (x_m * ((2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))) + (math.pow(x_m, 6.0) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0)))))) / math.sqrt(math.pi)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.002)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x_m ^ 2.0))))) / sqrt(pi)));
	else
		tmp = Float64(Float64(x_m * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + Float64((x_m ^ 6.0) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))))) / sqrt(pi));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.002)
		tmp = x_m * ((2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m ^ 2.0))))) / sqrt(pi));
	else
		tmp = (x_m * ((2.0 + (0.6666666666666666 * (x_m ^ 2.0))) + ((x_m ^ 6.0) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0)))))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.002], N[(x$95$m * N[(N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   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 99.9%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. *-commutative 93.1%

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

   7. Simplified 93.1%

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

      1. expm1-log1p-u 93.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)\right)} \]

      2. expm1-undefine 34.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)} - 1} \]

   9. Applied egg-rr 4.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 4.0%

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

       2. rem-exp-log 4.0%

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

       3. +-commutative 4.0%

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

       4. associate--l+ 29.8%

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

       5. metadata-eval 29.8%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

       6. metadata-eval 29.8%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

       7. sub-neg 29.8%

          \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

       8. --rgt-identity 29.8%

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

   11. Simplified 29.8%

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

   12. Taylor expanded in x around 0 29.8%

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

   if 2e-3 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

       1. fma-undefine 99.2%

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

   11. Applied egg-rr 99.2%

       \[\leadsto \frac{x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + {x}^{6} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;x \cdot \frac{2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(2 + 0.6666666666666666 \cdot {x}^{2}\right) + {x}^{6} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x\_m}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x\_m}^{2}} + \frac{0.6666666666666666}{{x\_m}^{4}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (*
    x_m
    (/
     (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (pow x_m 2.0)))))
     (sqrt PI)))
   (*
    (pow x_m 7.0)
    (*
     (sqrt (/ 1.0 PI))
     (+
      0.047619047619047616
      (+ (/ 0.2 (pow x_m 2.0)) (/ 0.6666666666666666 (pow x_m 4.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * ((2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * pow(x_m, 2.0))))) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + ((0.2 / pow(x_m, 2.0)) + (0.6666666666666666 / pow(x_m, 4.0)))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * ((2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x_m, 2.0))))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 + ((0.2 / Math.pow(x_m, 2.0)) + (0.6666666666666666 / Math.pow(x_m, 4.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = x_m * ((2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x_m, 2.0))))) / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (math.sqrt((1.0 / math.pi)) * (0.047619047619047616 + ((0.2 / math.pow(x_m, 2.0)) + (0.6666666666666666 / math.pow(x_m, 4.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x_m ^ 2.0))))) / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(Float64(0.2 / (x_m ^ 2.0)) + Float64(0.6666666666666666 / (x_m ^ 4.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.86)
		tmp = x_m * ((2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m ^ 2.0))))) / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (sqrt((1.0 / pi)) * (0.047619047619047616 + ((0.2 / (x_m ^ 2.0)) + (0.6666666666666666 / (x_m ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.86], N[(x$95$m * N[(N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 93.0%

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

      1. *-commutative 93.0%

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

   7. Simplified 93.0%

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

      1. expm1-log1p-u 93.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)\right)} \]

      2. expm1-undefine 35.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)} - 1} \]

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 4.5%

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

       2. rem-exp-log 4.6%

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

       3. +-commutative 4.6%

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

       4. associate--l+ 30.1%

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

       5. metadata-eval 30.1%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

       6. metadata-eval 30.1%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

       7. sub-neg 30.1%

          \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

       8. --rgt-identity 30.1%

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

   11. Simplified 30.1%

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

   12. Taylor expanded in x around 0 30.1%

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

   if 1.8600000000000001 < 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%

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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around inf 0.7%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.7%

         \[\leadsto {x}^{7} \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616} + \left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      2. associate-*r* 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \left(\color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      3. associate-*r* 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \left(\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right) \]

      4. distribute-rgt-out 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot \frac{1}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)}\right) \]

      5. distribute-lft-out 0.7%

         \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(0.2 \cdot \frac{1}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right)} \]

      6. associate-*r/ 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \]

      7. metadata-eval 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{\color{blue}{0.2}}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \]

      8. associate-*r/ 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \color{blue}{\frac{0.6666666666666666 \cdot 1}{{x}^{4}}}\right)\right)\right) \]

      9. metadata-eval 0.7%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{\color{blue}{0.6666666666666666}}{{x}^{4}}\right)\right)\right) \]

   9. Simplified 0.7%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;x\_m \cdot \frac{2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x\_m}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (*
    x_m
    (/
     (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (pow x_m 2.0)))))
     (sqrt PI)))
   (*
    (pow x_m 7.0)
    (* (sqrt (/ 1.0 PI)) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * ((2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * pow(x_m, 2.0))))) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * ((2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x_m, 2.0))))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = x_m * ((2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x_m, 2.0))))) / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (math.sqrt((1.0 / math.pi)) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x_m ^ 2.0))))) / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = x_m * ((2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m ^ 2.0))))) / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (sqrt((1.0 / pi)) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(x$95$m * N[(N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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 93.0%

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

      1. *-commutative 93.0%

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

   7. Simplified 93.0%

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

      1. expm1-log1p-u 93.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)\right)} \]

      2. expm1-undefine 35.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{{x}^{4} \cdot 0.2 + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)} - 1} \]

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 4.5%

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

       2. rem-exp-log 4.6%

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

       3. +-commutative 4.6%

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

       4. associate--l+ 30.1%

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

       5. metadata-eval 30.1%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

       6. metadata-eval 30.1%

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

       7. sub-neg 30.1%

          \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

       8. --rgt-identity 30.1%

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

   11. Simplified 30.1%

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

   12. Taylor expanded in x around 0 30.1%

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

   if 2.2000000000000002 < 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%

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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around inf 1.4%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 1.4%

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

      2. distribute-rgt-out 1.4%

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

      3. associate-*r/ 1.4%

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

      4. metadata-eval 1.4%

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

   9. Simplified 1.4%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \frac{2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (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 <= 1.86) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} 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 <= 1.86) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} 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 <= 1.86:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	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 <= 1.86)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	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 <= 1.86)
		tmp = x_m * (2.0 / sqrt(pi));
	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, 1.86], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $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 < 1.8600000000000001

   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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

   9. Simplified 29.7%

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

       1. associate-*l* 29.7%

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

       2. sqrt-div 29.7%

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

       3. metadata-eval 29.7%

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

       4. div-inv 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{x}{\sqrt{\pi}}} \]

       5. clear-num 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x}}} \]

       6. add-sqr-sqrt 28.0%

          \[\leadsto 2 \cdot \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       7. fabs-sqr 28.0%

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

       8. add-sqr-sqrt 64.5%

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

       9. un-div-inv 64.5%

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

       10. add-sqr-sqrt 28.0%

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

       11. fabs-sqr 28.0%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       12. add-sqr-sqrt 29.5%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{x}}} \]

   11. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 29.7%

          \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   if 1.8600000000000001 < 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%

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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around inf 3.6%

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

      1. associate-*r* 3.6%

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

      2. *-commutative 3.6%

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

   9. Simplified 3.6%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x\_m}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* x_m (/ 2.0 (sqrt 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 <= 1.86) {
		tmp = x_m * (2.0 / sqrt(((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 <= 1.86) {
		tmp = x_m * (2.0 / Math.sqrt(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 <= 1.86:
		tmp = x_m * (2.0 / math.sqrt(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 <= 1.86)
		tmp = Float64(x_m * Float64(2.0 / sqrt(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 <= 1.86)
		tmp = x_m * (2.0 / sqrt(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, 1.86], N[(x$95$m * N[(2.0 / N[Sqrt[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 < 1.8600000000000001

   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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

   9. Simplified 29.7%

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

       1. associate-*l* 29.7%

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

       2. sqrt-div 29.7%

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

       3. metadata-eval 29.7%

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

       4. div-inv 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{x}{\sqrt{\pi}}} \]

       5. clear-num 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x}}} \]

       6. add-sqr-sqrt 28.0%

          \[\leadsto 2 \cdot \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       7. fabs-sqr 28.0%

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

       8. add-sqr-sqrt 64.5%

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

       9. un-div-inv 64.5%

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

       10. add-sqr-sqrt 28.0%

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

       11. fabs-sqr 28.0%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       12. add-sqr-sqrt 29.5%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{x}}} \]

   11. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 29.7%

          \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   if 1.8600000000000001 < 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%

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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around inf 1.4%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 1.4%

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

      2. distribute-rgt-out 1.4%

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

      3. associate-*r/ 1.4%

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

      4. metadata-eval 1.4%

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

   9. Simplified 1.4%

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

   10. Taylor expanded in x around inf 3.6%

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

       1. associate-*r* 3.6%

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

   12. Simplified 3.6%

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

       1. add-sqr-sqrt 3.4%

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

       2. sqrt-unprod 36.5%

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

       3. *-commutative 36.5%

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

       4. *-commutative 36.5%

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

       5. swap-sqr 36.5%

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

       6. add-sqr-sqrt 36.5%

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

       7. *-commutative 36.5%

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

       8. *-commutative 36.5%

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

       9. swap-sqr 36.5%

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

       10. pow-prod-up 36.5%

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

       11. metadata-eval 36.5%

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

       12. metadata-eval 36.5%

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

   14. Applied egg-rr 36.5%

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

       1. associate-*r* 36.5%

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

       2. *-commutative 36.5%

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

       3. associate-*l/ 36.5%

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

       4. *-lft-identity 36.5%

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

   16. Simplified 36.5%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (pow x_m 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.86)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.86], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

   9. Simplified 29.7%

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

       1. associate-*l* 29.7%

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

       2. sqrt-div 29.7%

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

       3. metadata-eval 29.7%

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

       4. div-inv 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{x}{\sqrt{\pi}}} \]

       5. clear-num 29.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x}}} \]

       6. add-sqr-sqrt 28.0%

          \[\leadsto 2 \cdot \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       7. fabs-sqr 28.0%

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

       8. add-sqr-sqrt 64.5%

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

       9. un-div-inv 64.5%

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

       10. add-sqr-sqrt 28.0%

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

       11. fabs-sqr 28.0%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       12. add-sqr-sqrt 29.5%

           \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{x}}} \]

   11. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 29.7%

          \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

   if 1.8600000000000001 < 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%

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

   6. Applied egg-rr 30.1%

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

   7. Taylor expanded in x around inf 1.4%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 1.4%

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

      2. distribute-rgt-out 1.4%

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

      3. associate-*r/ 1.4%

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

      4. metadata-eval 1.4%

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

   9. Simplified 1.4%

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

   10. Taylor expanded in x around inf 3.6%

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

       1. associate-*r* 3.6%

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

   12. Simplified 3.6%

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

       1. sqrt-div 3.6%

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

       2. metadata-eval 3.6%

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

       3. un-div-inv 3.6%

          \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]

   14. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
   15. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \frac{\color{blue}{{x}^{7} \cdot 0.047619047619047616}}{\sqrt{\pi}} \]

       2. associate-/l* 3.6%

          \[\leadsto \color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}} \]

   16. Simplified 3.6%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.4% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

6. Applied egg-rr 30.1%

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

7. Taylor expanded in x around inf 14.4%

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

   1. associate-*r/ 14.4%

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

   2. metadata-eval 14.4%

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

9. Simplified 14.4%

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

10. Taylor expanded in x around 0 13.8%

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

11. Taylor expanded in x around inf 29.4%

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

12. Final simplification 29.4%

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

Alternative 13: 67.9% accurate, 17.6× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (2.0 / sqrt(((double) M_PI)));
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (2.0 / math.sqrt(math.pi))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(2.0 / N[Sqrt[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%

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

6. Applied egg-rr 30.1%

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

7. Taylor expanded in x around 0 29.7%

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

   1. associate-*r* 29.7%

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

9. Simplified 29.7%

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

    1. associate-*l* 29.7%

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

    2. sqrt-div 29.7%

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

    3. metadata-eval 29.7%

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

    4. div-inv 29.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{x}{\sqrt{\pi}}} \]

    5. clear-num 29.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x}}} \]

    6. add-sqr-sqrt 28.0%

       \[\leadsto 2 \cdot \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

    7. fabs-sqr 28.0%

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

    8. add-sqr-sqrt 64.5%

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

    9. un-div-inv 64.5%

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

    10. add-sqr-sqrt 28.0%

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

    11. fabs-sqr 28.0%

        \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

    12. add-sqr-sqrt 29.5%

        \[\leadsto \frac{2}{\frac{\sqrt{\pi}}{\color{blue}{x}}} \]

11. Applied egg-rr 29.5%

    \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{\pi}}{x}}} \]
12. Step-by-step derivation

    1. associate-/r/ 29.7%

       \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

13. Simplified 29.7%

    \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot x} \]

14. Final simplification 29.7%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
15. Add Preprocessing

Alternative 14: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

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

6. Applied egg-rr 30.1%

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

7. Taylor expanded in x around 0 29.7%

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

   1. associate-*r* 29.7%

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

9. Simplified 29.7%

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

    1. expm1-log1p-u 29.6%

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

    2. expm1-undefine 4.1%

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

11. Applied egg-rr 4.1%

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

12. Taylor expanded in x around 0 4.1%

    \[\leadsto \color{blue}{1} - 1 \]

13. Final simplification 4.1%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))))))