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

Percentage Accurate: 99.8% → 99.8%

Time: 18.0s

Alternatives: 11

Speedup: 3.0×

• 29.3% 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, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\_m\right|, 0.6666666666666666 \cdot {x\_m}^{3}\right) + 0.2 \cdot {x\_m}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left({x\_m}^{3} \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (fma 2.0 (fabs x_m) (* 0.6666666666666666 (pow x_m 3.0)))
     (* 0.2 (pow x_m 5.0)))
    (* 0.047619047619047616 (* (* x_m x_m) (* (pow x_m 3.0) (* x_m x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((fma(2.0, fabs(x_m), (0.6666666666666666 * pow(x_m, 3.0))) + (0.2 * pow(x_m, 5.0))) + (0.047619047619047616 * ((x_m * x_m) * (pow(x_m, 3.0) * (x_m * x_m)))))));
}

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x_m), Float64(0.6666666666666666 * (x_m ^ 3.0))) + Float64(0.2 * (x_m ^ 5.0))) + Float64(0.047619047619047616 * Float64(Float64(x_m * x_m) * Float64((x_m ^ 3.0) * Float64(x_m * x_m)))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x$95$m], $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. associate-*r* 99.9%

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

   2. rem-square-sqrt 34.3%

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

   3. fabs-sqr 34.3%

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

   4. rem-square-sqrt 78.2%

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

   5. associate-*l* 78.2%

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

   6. pow-plus 78.2%

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

   7. metadata-eval 78.2%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot {x}^{\color{blue}{5}}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Simplified 78.2%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.2 \cdot {x}^{5}}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Taylor expanded in x around 0 78.2%

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

   1. associate-*r* 78.2%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. rem-square-sqrt 34.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. fabs-sqr 34.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. rem-square-sqrt 78.1%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   5. associate-*r* 78.1%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \color{blue}{0.6666666666666666 \cdot \left({x}^{2} \cdot x\right)}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. unpow2 78.1%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. unpow3 78.1%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

10. Simplified 78.1%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \color{blue}{0.6666666666666666 \cdot {x}^{3}}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Taylor expanded in x around 0 78.1%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. unpow2 78.1%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    2. rem-square-sqrt 34.3%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    3. fabs-sqr 34.3%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    4. rem-square-sqrt 99.3%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    5. unpow3 99.3%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

13. Simplified 99.3%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

14. Final simplification 99.3%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{3} \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
15. Add Preprocessing

Alternative 2: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 34.0%

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

   2. fabs-sqr 34.0%

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

   3. add-sqr-sqrt 99.4%

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

   4. div-inv 99.8%

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

   5. pow1/2 99.8%

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

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 35.6%

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

7. Final simplification 35.6%

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

Alternative 5: 99.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 98.9%

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

   1. log1p-expm1-u 94.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   2. log1p-undefine 28.1%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   3. add-sqr-sqrt 3.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right) \]

   4. fabs-sqr 3.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right) \]

   5. add-sqr-sqrt 5.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{x} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right) \]

   6. add-sqr-sqrt 5.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}}}\right|\right)\right) \]

   7. fabs-sqr 5.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}}\right)}\right)\right) \]

   8. add-sqr-sqrt 5.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}}\right)\right) \]

7. Applied egg-rr 5.3%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot \frac{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}{\sqrt{\pi}}\right)\right)} \]
8. Step-by-step derivation

   1. log1p-define 35.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}{\sqrt{\pi}}\right)\right)} \]

   2. log1p-expm1-u 35.3%

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

   3. *-commutative 35.3%

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

   4. fma-define 35.3%

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

9. Applied egg-rr 35.3%

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

10. Final simplification 35.3%

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

Alternative 6: 98.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 98.9%

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

   1. add-sqr-sqrt 33.7%

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

   2. fabs-sqr 33.7%

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

   3. add-sqr-sqrt 33.3%

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

   4. fabs-sqr 33.3%

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

   5. add-sqr-sqrt 33.7%

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

   6. add-sqr-sqrt 35.3%

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

   7. clear-num 35.3%

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

   8. un-div-inv 35.2%

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

   9. fma-undefine 35.2%

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

7. Applied egg-rr 35.2%

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

8. Final simplification 35.2%

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

Alternative 7: 98.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.35:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x\_m \cdot \left(2 + 0.2 \cdot {x\_m}^{4}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x\_m}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.35)
   (fabs (* (sqrt (/ 1.0 PI)) (* x_m (+ 2.0 (* 0.2 (pow x_m 4.0))))))
   (fabs (* 0.047619047619047616 (/ (pow x_m 7.0) (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.35) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x_m * (2.0 + (0.2 * pow(x_m, 4.0))))));
	} else {
		tmp = fabs((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 (x_m <= 2.35) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x_m * (2.0 + (0.2 * Math.pow(x_m, 4.0))))));
	} else {
		tmp = Math.abs((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 x_m <= 2.35:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x_m * (2.0 + (0.2 * math.pow(x_m, 4.0))))))
	else:
		tmp = math.fabs((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 (x_m <= 2.35)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x_m * Float64(2.0 + Float64(0.2 * (x_m ^ 4.0))))));
	else
		tmp = abs(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 (x_m <= 2.35)
		tmp = abs((sqrt((1.0 / pi)) * (x_m * (2.0 + (0.2 * (x_m ^ 4.0))))));
	else
		tmp = abs((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[x$95$m, 2.35], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.35000000000000009

   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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around 0 93.9%

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

   if 2.35000000000000009 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around inf 30.5%

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

      1. associate-*r* 30.5%

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

      2. sqrt-div 30.5%

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

      3. metadata-eval 30.5%

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

      4. un-div-inv 30.5%

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

   10. Applied egg-rr 30.5%

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

       1. associate-/l* 30.5%

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

   12. Simplified 30.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.2 \cdot {x}^{4}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.72:\\ \;\;\;\;\left|x\_m \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x\_m}^{6} \cdot 0.4444444444444444}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.72)
   (fabs (* x_m (/ 2.0 (sqrt PI))))
   (fabs (sqrt (/ (* (pow x_m 6.0) 0.4444444444444444) PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.72) {
		tmp = fabs((x_m * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt(((pow(x_m, 6.0) * 0.4444444444444444) / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.72) {
		tmp = Math.abs((x_m * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt(((Math.pow(x_m, 6.0) * 0.4444444444444444) / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.72:
		tmp = math.fabs((x_m * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt(((math.pow(x_m, 6.0) * 0.4444444444444444) / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.72)
		tmp = abs(Float64(x_m * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(Float64((x_m ^ 6.0) * 0.4444444444444444) / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.72)
		tmp = abs((x_m * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt((((x_m ^ 6.0) * 0.4444444444444444) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.72], N[Abs[N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.4444444444444444), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.71999999999999997

   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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

   10. Simplified 73.9%

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

       1. sqrt-div 73.9%

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

       2. metadata-eval 73.9%

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

       3. un-div-inv 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. associate-/l* 73.9%

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

   14. Applied egg-rr 73.9%

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

   if 1.71999999999999997 < 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 99.4%

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

   5. Taylor expanded in x around inf 22.7%

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

      1. *-commutative 22.7%

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

      2. *-commutative 22.7%

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

      3. associate-*l* 22.7%

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

      4. associate-*l* 22.7%

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

      5. rem-square-sqrt 2.2%

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

      6. fabs-sqr 2.2%

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

      7. rem-square-sqrt 22.7%

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

      8. associate-*l* 22.7%

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

      9. unpow2 22.7%

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

      10. unpow3 22.7%

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

      11. *-commutative 22.7%

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

   7. Simplified 22.7%

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

      1. add-sqr-sqrt 3.6%

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

      2. sqrt-unprod 26.6%

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

      3. swap-sqr 26.6%

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

      4. add-sqr-sqrt 26.6%

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

      5. *-commutative 26.6%

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

      6. *-commutative 26.6%

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

      7. swap-sqr 26.6%

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

      8. pow-sqr 26.6%

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

      9. metadata-eval 26.6%

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

      10. metadata-eval 26.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left({x}^{6} \cdot \color{blue}{0.4444444444444444}\right)}\right| \]

   9. Applied egg-rr 26.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{6} \cdot 0.4444444444444444\right)}}\right| \]
   10. Step-by-step derivation

       1. associate-*l/ 26.6%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{1 \cdot \left({x}^{6} \cdot 0.4444444444444444\right)}{\pi}}}\right| \]

       2. *-lft-identity 26.6%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{6} \cdot 0.4444444444444444}}{\pi}}\right| \]

   11. Simplified 26.6%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{6} \cdot 0.4444444444444444}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{6} \cdot 0.4444444444444444}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;\left|x\_m \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x\_m}^{14}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (fabs (* x_m (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (sqrt (/ (pow x_m 14.0) PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = fabs((x_m * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * sqrt((pow(x_m, 14.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = Math.abs((x_m * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.sqrt((Math.pow(x_m, 14.0) / Math.PI))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = math.fabs((x_m * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * math.sqrt((math.pow(x_m, 14.0) / math.pi))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = abs(Float64(x_m * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * sqrt(Float64((x_m ^ 14.0) / pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.86)
		tmp = abs((x_m * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * sqrt(((x_m ^ 14.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.86], N[Abs[N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[Sqrt[N[(N[Power[x$95$m, 14.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

   10. Simplified 73.9%

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

       1. sqrt-div 73.9%

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

       2. metadata-eval 73.9%

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

       3. un-div-inv 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. associate-/l* 73.9%

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

   14. Applied egg-rr 73.9%

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

   if 1.8600000000000001 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around inf 30.5%

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

      1. add-sqr-sqrt 3.8%

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

      2. sqrt-unprod 29.0%

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

      3. *-commutative 29.0%

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

      4. *-commutative 29.0%

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

      5. swap-sqr 29.0%

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

      6. add-sqr-sqrt 29.0%

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

      7. pow-prod-up 29.0%

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

      8. metadata-eval 29.0%

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

   10. Applied egg-rr 29.0%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot {x}^{14}}}\right| \]
   11. Step-by-step derivation

       1. associate-*l/ 29.0%

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

       2. *-lft-identity 29.0%

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

   12. Simplified 29.0%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{{x}^{14}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = fabs((x_m * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((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 (x_m <= 1.86) {
		tmp = Math.abs((x_m * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((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 x_m <= 1.86:
		tmp = math.fabs((x_m * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((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 (x_m <= 1.86)
		tmp = abs(Float64(x_m * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(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 (x_m <= 1.86)
		tmp = abs((x_m * (2.0 / sqrt(pi))));
	else
		tmp = abs((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[x$95$m, 1.86], N[Abs[N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

   10. Simplified 73.9%

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

       1. sqrt-div 73.9%

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

       2. metadata-eval 73.9%

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

       3. un-div-inv 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. associate-/l* 73.9%

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

   14. Applied egg-rr 73.9%

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

   if 1.8600000000000001 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.7%

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

      5. fabs-sqr 33.7%

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

      6. rem-square-sqrt 77.6%

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

      7. +-commutative 77.6%

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

      8. fma-define 77.6%

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

      9. rem-square-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. rem-square-sqrt 77.6%

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

      12. rem-square-sqrt 33.9%

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

      13. fabs-sqr 33.9%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.9%

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

      17. fabs-sqr 33.9%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

   8. Taylor expanded in x around inf 30.5%

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

      1. associate-*r* 30.5%

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

      2. sqrt-div 30.5%

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

      3. metadata-eval 30.5%

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

      4. un-div-inv 30.5%

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

   10. Applied egg-rr 30.5%

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

       1. associate-/l* 30.5%

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

   12. Simplified 30.5%

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

4. Final simplification 73.9%

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

Alternative 11: 67.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.9%

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

   1. associate-+r+ 98.9%

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

   2. distribute-lft-in 98.9%

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

   3. fma-define 98.9%

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

   4. rem-square-sqrt 33.7%

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

   5. fabs-sqr 33.7%

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

   6. rem-square-sqrt 77.6%

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

   7. +-commutative 77.6%

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

   8. fma-define 77.6%

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

   9. rem-square-sqrt 33.9%

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

   10. fabs-sqr 33.9%

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

   11. rem-square-sqrt 77.6%

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

   12. rem-square-sqrt 33.9%

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

   13. fabs-sqr 33.9%

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

   14. rem-square-sqrt 98.9%

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

   15. *-commutative 98.9%

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

   16. rem-square-sqrt 33.9%

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

   17. fabs-sqr 33.9%

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

   18. rem-square-sqrt 98.9%

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

7. Simplified 98.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), 0.2 \cdot {x}^{5}\right)}\right| \]

8. Taylor expanded in x around 0 73.9%

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

   1. associate-*r* 73.9%

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

   2. *-commutative 73.9%

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

10. Simplified 73.9%

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

    1. sqrt-div 73.9%

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

    2. metadata-eval 73.9%

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

    3. un-div-inv 73.4%

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

12. Applied egg-rr 73.4%

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

    1. associate-/l* 73.9%

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

14. Applied egg-rr 73.9%

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

15. Final simplification 73.9%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))))))