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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

Alternatives: 11

Speedup: 3.2×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. +-commutative 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

   7. +-commutative 99.4%

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

6. Simplified 99.4%

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

   1. add-sqr-sqrt 32.5%

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

   2. fabs-sqr 32.5%

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

   3. add-sqr-sqrt 33.9%

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

   4. clear-num 33.9%

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

   5. associate-/r/ 34.1%

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

   6. clear-num 34.1%

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

8. Applied egg-rr 34.1%

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

   1. fma-udef 34.1%

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

10. Applied egg-rr 34.1%

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

11. Final simplification 34.1%

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

Alternative 2: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;x_m \cdot \left(\left(0.2 \cdot {x_m}^{4} + \mathsf{fma}\left(0.6666666666666666, {x_m}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{\mathsf{fma}\left(0.2, {x_m}^{4}, 0.047619047619047616 \cdot {x_m}^{6}\right) + 0.6666666666666666 \cdot {x_m}^{2}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (*
    x_m
    (*
     (+ (* 0.2 (pow x_m 4.0)) (fma 0.6666666666666666 (pow x_m 2.0) 2.0))
     (pow PI -0.5)))
   (*
    x_m
    (/
     (+
      (fma 0.2 (pow x_m 4.0) (* 0.047619047619047616 (pow x_m 6.0)))
      (* 0.6666666666666666 (pow x_m 2.0)))
     (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = x_m * (((0.2 * pow(x_m, 4.0)) + fma(0.6666666666666666, pow(x_m, 2.0), 2.0)) * pow(((double) M_PI), -0.5));
	} else {
		tmp = x_m * ((fma(0.2, pow(x_m, 4.0), (0.047619047619047616 * pow(x_m, 6.0))) + (0.6666666666666666 * pow(x_m, 2.0))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64(x_m * Float64(Float64(Float64(0.2 * (x_m ^ 4.0)) + fma(0.6666666666666666, (x_m ^ 2.0), 2.0)) * (pi ^ -0.5)));
	else
		tmp = Float64(x_m * Float64(Float64(fma(0.2, (x_m ^ 4.0), Float64(0.047619047619047616 * (x_m ^ 6.0))) + Float64(0.6666666666666666 * (x_m ^ 2.0))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(x$95$m * N[(N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(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]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. add-sqr-sqrt 50.1%

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

      2. fabs-sqr 50.1%

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

      3. add-sqr-sqrt 52.3%

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

      4. div-inv 52.3%

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

      5. *-un-lft-identity 52.3%

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

      6. times-frac 52.6%

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

      7. pow1/2 52.6%

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

      8. pow-flip 52.6%

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

      9. metadata-eval 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-/r/ 52.6%

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

      3. /-rgt-identity 52.6%

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

      4. associate-*l* 52.6%

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

   10. Simplified 52.6%

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

   11. Taylor expanded in x around 0 52.6%

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

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;x \cdot \left(\left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 0.6666666666666666 \cdot {x}^{2}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 3: 99.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;x_m \cdot \left(\left(0.2 \cdot {x_m}^{4} + \mathsf{fma}\left(0.6666666666666666, {x_m}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (*
    x_m
    (*
     (+ (* 0.2 (pow x_m 4.0)) (fma 0.6666666666666666 (pow x_m 2.0) 2.0))
     (pow PI -0.5)))
   (*
    (sqrt (/ 1.0 PI))
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = x_m * (((0.2 * pow(x_m, 4.0)) + fma(0.6666666666666666, pow(x_m, 2.0), 2.0)) * pow(((double) M_PI), -0.5));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64(x_m * Float64(Float64(Float64(0.2 * (x_m ^ 4.0)) + fma(0.6666666666666666, (x_m ^ 2.0), 2.0)) * (pi ^ -0.5)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(x$95$m * N[(N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. add-sqr-sqrt 50.1%

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

      2. fabs-sqr 50.1%

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

      3. add-sqr-sqrt 52.3%

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

      4. div-inv 52.3%

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

      5. *-un-lft-identity 52.3%

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

      6. times-frac 52.6%

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

      7. pow1/2 52.6%

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

      8. pow-flip 52.6%

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

      9. metadata-eval 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-/r/ 52.6%

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

      3. /-rgt-identity 52.6%

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

      4. associate-*l* 52.6%

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

   10. Simplified 52.6%

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

   11. Taylor expanded in x around 0 52.6%

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

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 0.1%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. *-commutative 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)} \cdot \sqrt{\frac{1}{\pi}} \]

       5. distribute-rgt-out 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + {x}^{7} \cdot 0.047619047619047616\right)} \]

       6. *-commutative 0.1%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \color{blue}{0.047619047619047616 \cdot {x}^{7}}\right) \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;x \cdot \left(\left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;x_m \cdot \frac{\left(2 + 0.6666666666666666 \cdot {x_m}^{2}\right) + 0.2 \cdot {x_m}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (*
    x_m
    (/
     (+ (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))) (* 0.2 (pow x_m 4.0)))
     (sqrt PI)))
   (*
    (sqrt (/ 1.0 PI))
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = x_m * (((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) + (0.2 * pow(x_m, 4.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x_m, 5.0)) + (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 (Math.abs(x_m) <= 0.004) {
		tmp = x_m * (((2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))) + (0.2 * Math.pow(x_m, 4.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x_m, 5.0)) + (0.047619047619047616 * Math.pow(x_m, 7.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = x_m * (((2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))) + (0.2 * math.pow(x_m, 4.0))) / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x_m, 5.0)) + (0.047619047619047616 * math.pow(x_m, 7.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64(x_m * Float64(Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + Float64(0.2 * (x_m ^ 4.0))) / sqrt(pi)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x_m ^ 5.0)) + 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 (abs(x_m) <= 0.004)
		tmp = x_m * (((2.0 + (0.6666666666666666 * (x_m ^ 2.0))) + (0.2 * (x_m ^ 4.0))) / sqrt(pi));
	else
		tmp = sqrt((1.0 / pi)) * ((0.2 * (x_m ^ 5.0)) + (0.047619047619047616 * (x_m ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(x$95$m * N[(N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. add-sqr-sqrt 50.1%

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

      2. fabs-sqr 50.1%

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

      3. add-sqr-sqrt 52.3%

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

      4. clear-num 52.3%

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

      5. associate-/r/ 52.6%

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

      6. clear-num 52.6%

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

   8. Applied egg-rr 52.6%

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

   9. Taylor expanded in x around 0 52.6%

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

       1. fma-udef 52.6%

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

   11. Applied egg-rr 52.6%

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

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 0.1%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. *-commutative 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)} \cdot \sqrt{\frac{1}{\pi}} \]

       5. distribute-rgt-out 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + {x}^{7} \cdot 0.047619047619047616\right)} \]

       6. *-commutative 0.1%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \color{blue}{0.047619047619047616 \cdot {x}^{7}}\right) \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;x \cdot \frac{\left(2 + 0.6666666666666666 \cdot {x}^{2}\right) + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{{x_m}^{2} \cdot -0.16666666666666666 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (* (pow PI -0.5) (/ x_m (+ (* (pow x_m 2.0) -0.16666666666666666) 0.5)))
   (*
    (sqrt (/ 1.0 PI))
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / ((pow(x_m, 2.0) * -0.16666666666666666) + 0.5));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x_m, 5.0)) + (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 (Math.abs(x_m) <= 0.004) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m / ((Math.pow(x_m, 2.0) * -0.16666666666666666) + 0.5));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x_m, 5.0)) + (0.047619047619047616 * Math.pow(x_m, 7.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = math.pow(math.pi, -0.5) * (x_m / ((math.pow(x_m, 2.0) * -0.16666666666666666) + 0.5))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x_m, 5.0)) + (0.047619047619047616 * math.pow(x_m, 7.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / Float64(Float64((x_m ^ 2.0) * -0.16666666666666666) + 0.5)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x_m ^ 5.0)) + 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 (abs(x_m) <= 0.004)
		tmp = (pi ^ -0.5) * (x_m / (((x_m ^ 2.0) * -0.16666666666666666) + 0.5));
	else
		tmp = sqrt((1.0 / pi)) * ((0.2 * (x_m ^ 5.0)) + (0.047619047619047616 * (x_m ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{\left|x\right|}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 50.0%

          \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       2. fabs-sqr 50.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       3. add-sqr-sqrt 52.2%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       4. *-un-lft-identity 52.2%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       5. times-frac 52.5%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}} \]

       6. pow1/2 52.5%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       7. pow-flip 52.5%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       8. metadata-eval 52.5%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       9. +-commutative 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

       10. fma-def 52.5%

           \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 52.5%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   12. Step-by-step derivation

       1. fma-udef 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

   13. Applied egg-rr 52.5%

       \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 0.1%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. *-commutative 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)} \cdot \sqrt{\frac{1}{\pi}} \]

       5. distribute-rgt-out 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + {x}^{7} \cdot 0.047619047619047616\right)} \]

       6. *-commutative 0.1%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \color{blue}{0.047619047619047616 \cdot {x}^{7}}\right) \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 6: 98.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{{x_m}^{2} \cdot -0.16666666666666666 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (* (pow PI -0.5) (/ x_m (+ (* (pow x_m 2.0) -0.16666666666666666) 0.5)))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / ((pow(x_m, 2.0) * -0.16666666666666666) + 0.5));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.004) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m / ((Math.pow(x_m, 2.0) * -0.16666666666666666) + 0.5));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = math.pow(math.pi, -0.5) * (x_m / ((math.pow(x_m, 2.0) * -0.16666666666666666) + 0.5))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / Float64(Float64((x_m ^ 2.0) * -0.16666666666666666) + 0.5)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.004)
		tmp = (pi ^ -0.5) * (x_m / (((x_m ^ 2.0) * -0.16666666666666666) + 0.5));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{\left|x\right|}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 50.0%

          \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       2. fabs-sqr 50.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       3. add-sqr-sqrt 52.2%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       4. *-un-lft-identity 52.2%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       5. times-frac 52.5%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}} \]

       6. pow1/2 52.5%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       7. pow-flip 52.5%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       8. metadata-eval 52.5%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       9. +-commutative 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

       10. fma-def 52.5%

           \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 52.5%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   12. Step-by-step derivation

       1. fma-udef 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

   13. Applied egg-rr 52.5%

       \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. pow2 0.0%

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

       3. sqrt-prod 0.0%

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

       4. sqrt-pow1 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{\color{blue}{3.5}} \cdot \sqrt{\sqrt{\frac{1}{\pi}}}\right)}^{2} \]

       6. inv-pow 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\sqrt{\color{blue}{{\pi}^{-1}}}}\right)}^{2} \]

       7. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right)}^{2} \]

       8. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{{\pi}^{\color{blue}{-0.5}}}\right)}^{2} \]

       9. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)}}\right)}^{2} \]

       10. metadata-eval 0.0%

           \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot {\pi}^{\color{blue}{-0.25}}\right)}^{2} \]

   11. Applied egg-rr 0.0%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{{\left({x}^{3.5} \cdot {\pi}^{-0.25}\right)}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {\pi}^{-0.25}\right) \cdot \left({x}^{3.5} \cdot {\pi}^{-0.25}\right)\right)} \]

       2. swap-sqr 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {x}^{3.5}\right) \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right)} \]

       3. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left(\color{blue}{{x}^{\left(2 \cdot 3.5\right)}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       4. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{\color{blue}{7}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       5. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}\right) \]

       6. metadata-eval 0.1%

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

   13. Simplified 0.1%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 7: 98.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;x_m \cdot \frac{2 + 0.2 \cdot {x_m}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (* x_m (/ (+ 2.0 (* 0.2 (pow x_m 4.0))) (sqrt PI)))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = x_m * ((2.0 + (0.2 * pow(x_m, 4.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.004) {
		tmp = x_m * ((2.0 + (0.2 * Math.pow(x_m, 4.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = x_m * ((2.0 + (0.2 * math.pow(x_m, 4.0))) / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64(0.2 * (x_m ^ 4.0))) / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.004)
		tmp = x_m * ((2.0 + (0.2 * (x_m ^ 4.0))) / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(x$95$m * N[(N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. add-sqr-sqrt 50.1%

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

      2. fabs-sqr 50.1%

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

      3. add-sqr-sqrt 52.3%

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

      4. clear-num 52.3%

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

      5. associate-/r/ 52.6%

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

      6. clear-num 52.6%

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

   8. Applied egg-rr 52.6%

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

   9. Taylor expanded in x around 0 52.6%

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

   10. Taylor expanded in x around 0 52.3%

       \[\leadsto \frac{0.2 \cdot {x}^{4} + \color{blue}{2}}{\sqrt{\pi}} \cdot x \]

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. pow2 0.0%

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

       3. sqrt-prod 0.0%

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

       4. sqrt-pow1 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{\color{blue}{3.5}} \cdot \sqrt{\sqrt{\frac{1}{\pi}}}\right)}^{2} \]

       6. inv-pow 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\sqrt{\color{blue}{{\pi}^{-1}}}}\right)}^{2} \]

       7. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right)}^{2} \]

       8. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{{\pi}^{\color{blue}{-0.5}}}\right)}^{2} \]

       9. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)}}\right)}^{2} \]

       10. metadata-eval 0.0%

           \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot {\pi}^{\color{blue}{-0.25}}\right)}^{2} \]

   11. Applied egg-rr 0.0%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{{\left({x}^{3.5} \cdot {\pi}^{-0.25}\right)}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {\pi}^{-0.25}\right) \cdot \left({x}^{3.5} \cdot {\pi}^{-0.25}\right)\right)} \]

       2. swap-sqr 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {x}^{3.5}\right) \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right)} \]

       3. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left(\color{blue}{{x}^{\left(2 \cdot 3.5\right)}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       4. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{\color{blue}{7}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       5. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}\right) \]

       6. metadata-eval 0.1%

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

   13. Simplified 0.1%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 8: 98.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (* (pow PI -0.5) (/ x_m 0.5))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / 0.5);
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.004) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m / 0.5);
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = math.pow(math.pi, -0.5) * (x_m / 0.5)
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / 0.5));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.004)
		tmp = (pi ^ -0.5) * (x_m / 0.5);
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / 0.5), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{\left|x\right|}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 50.0%

          \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       2. fabs-sqr 50.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       3. add-sqr-sqrt 52.2%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       4. *-un-lft-identity 52.2%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       5. times-frac 52.5%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}} \]

       6. pow1/2 52.5%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       7. pow-flip 52.5%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       8. metadata-eval 52.5%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       9. +-commutative 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

       10. fma-def 52.5%

           \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 52.5%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   12. Taylor expanded in x around 0 52.3%

       \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{0.5}} \]

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. pow2 0.0%

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

       3. sqrt-prod 0.0%

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

       4. sqrt-pow1 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{\color{blue}{3.5}} \cdot \sqrt{\sqrt{\frac{1}{\pi}}}\right)}^{2} \]

       6. inv-pow 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\sqrt{\color{blue}{{\pi}^{-1}}}}\right)}^{2} \]

       7. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right)}^{2} \]

       8. metadata-eval 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \sqrt{{\pi}^{\color{blue}{-0.5}}}\right)}^{2} \]

       9. sqrt-pow1 0.0%

          \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot \color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)}}\right)}^{2} \]

       10. metadata-eval 0.0%

           \[\leadsto 0.047619047619047616 \cdot {\left({x}^{3.5} \cdot {\pi}^{\color{blue}{-0.25}}\right)}^{2} \]

   11. Applied egg-rr 0.0%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{{\left({x}^{3.5} \cdot {\pi}^{-0.25}\right)}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {\pi}^{-0.25}\right) \cdot \left({x}^{3.5} \cdot {\pi}^{-0.25}\right)\right)} \]

       2. swap-sqr 0.0%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{3.5} \cdot {x}^{3.5}\right) \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right)} \]

       3. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left(\color{blue}{{x}^{\left(2 \cdot 3.5\right)}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       4. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{\color{blue}{7}} \cdot \left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)\right) \]

       5. pow-sqr 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}\right) \]

       6. metadata-eval 0.1%

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

   13. Simplified 0.1%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]

Alternative 9: 98.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x_m}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.004)
   (* (pow PI -0.5) (/ x_m 0.5))
   (* 0.047619047619047616 (/ (pow x_m 7.0) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.004) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / 0.5);
	} else {
		tmp = 0.047619047619047616 * (pow(x_m, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.004) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m / 0.5);
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.004:
		tmp = math.pow(math.pi, -0.5) * (x_m / 0.5)
	else:
		tmp = 0.047619047619047616 * (math.pow(x_m, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.004)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / 0.5));
	else
		tmp = Float64(0.047619047619047616 * Float64((x_m ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.004)
		tmp = (pi ^ -0.5) * (x_m / 0.5);
	else
		tmp = 0.047619047619047616 * ((x_m ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.004], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / 0.5), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0040000000000000001

   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 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. rem-square-sqrt 99.5%

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

      5. fabs-sqr 99.5%

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

      6. rem-square-sqrt 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{\left|x\right|}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 99.1%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 99.1%

         \[\leadsto \frac{\left|x\right|}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 99.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 50.0%

          \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       2. fabs-sqr 50.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       3. add-sqr-sqrt 52.2%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       4. *-un-lft-identity 52.2%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

       5. times-frac 52.5%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}} \]

       6. pow1/2 52.5%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       7. pow-flip 52.5%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       8. metadata-eval 52.5%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

       9. +-commutative 52.5%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

       10. fma-def 52.5%

           \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 52.5%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   12. Taylor expanded in x around 0 52.3%

       \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{0.5}} \]

   if 0.0040000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. rem-square-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. clear-num 0.1%

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

      5. associate-/r/ 0.1%

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

      6. clear-num 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 0.1%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

       2. expm1-udef 0.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1} \]

       3. sqrt-div 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

       4. metadata-eval 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

       5. un-div-inv 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}}}\right)} - 1 \]

   11. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)} \]

       2. expm1-log1p 0.1%

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

   13. Simplified 0.1%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 10: 66.9% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in x around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. +-commutative 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

   7. +-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in x around 0 64.7%

   \[\leadsto \frac{\left|x\right|}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
8. Step-by-step derivation

   1. +-commutative 64.7%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

   2. associate-*r* 64.7%

      \[\leadsto \frac{\left|x\right|}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

   3. distribute-rgt-out 64.7%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

   4. *-commutative 64.7%

      \[\leadsto \frac{\left|x\right|}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

9. Simplified 64.7%

   \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 32.4%

       \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

    2. fabs-sqr 32.4%

       \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

    3. add-sqr-sqrt 34.7%

       \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

    4. *-un-lft-identity 34.7%

       \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)} \]

    5. times-frac 34.9%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}} \]

    6. pow1/2 34.9%

       \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

    7. pow-flip 34.9%

       \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

    8. metadata-eval 34.9%

       \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666} \]

    9. +-commutative 34.9%

       \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

    10. fma-def 34.9%

        \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

11. Applied egg-rr 34.9%

    \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

12. Taylor expanded in x around 0 34.0%

    \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{0.5}} \]

13. Final simplification 34.0%

    \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{0.5} \]

Alternative 11: 66.4% accurate, 9.5× speedup

Math

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

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(Float64(x_m * 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[(N[(x$95$m * 2.0), $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 99.4%

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

4. Taylor expanded in x around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. +-commutative 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

   7. +-commutative 99.4%

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

6. Simplified 99.4%

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

   1. add-sqr-sqrt 32.5%

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

   2. fabs-sqr 32.5%

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

   3. add-sqr-sqrt 33.9%

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

   4. clear-num 33.9%

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

   5. associate-/r/ 34.1%

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

   6. clear-num 34.1%

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

8. Applied egg-rr 34.1%

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

9. Taylor expanded in x around 0 34.0%

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

    1. associate-*r* 34.0%

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

11. Simplified 34.0%

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

    1. sqrt-div 34.0%

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

    2. metadata-eval 34.0%

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

    3. un-div-inv 33.9%

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

    4. *-commutative 33.9%

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

13. Applied egg-rr 33.9%

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

14. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2023333 
(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)))))))