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

Percentage Accurate: 99.8% → 99.9%

Time: 15.1s

Alternatives: 11

Speedup: 4.4×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. add0 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| + 0} \]

7. Applied egg-rr 37.5%

   \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 0} \]
8. Step-by-step derivation

   1. associate-*r* 37.5%

      \[\leadsto \color{blue}{\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)} + 0 \]

   2. add0 37.5%

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

   3. *-commutative 37.5%

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

   4. +-commutative 37.5%

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

   5. fma-undefine 37.5%

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

   6. associate-+l+ 37.5%

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

   7. fma-define 37.5%

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

   8. +-commutative 37.5%

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

   9. associate-+r+ 37.5%

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

   10. fma-define 37.5%

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

   11. fma-undefine 37.5%

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

9. Simplified 37.5%

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

    1. fma-undefine 37.5%

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

    2. fma-undefine 37.5%

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

    3. associate-+r+ 37.5%

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

11. Applied egg-rr 37.5%

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

12. Final simplification 37.5%

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

Alternative 2: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. add0 99.4%

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

      2. add-sqr-sqrt 51.3%

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

      3. fabs-sqr 51.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. add-sqr-sqrt 52.4%

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

      6. fabs-sqr 52.4%

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

      7. add-sqr-sqrt 53.3%

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

      8. +-commutative 53.3%

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

      9. fma-define 53.3%

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

      10. pow2 53.3%

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

   7. Applied egg-rr 53.3%

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

      1. add0 53.3%

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

   9. Simplified 53.3%

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

       1. fma-undefine 53.3%

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

   11. Applied egg-rr 53.3%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. add0 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}} + 0\right)}\right| \]

      2. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)} + 0\right)\right| \]

      3. inv-pow 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      4. sqrt-pow1 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      5. metadata-eval 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      6. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)} + 0\right)\right| \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right) + 0\right)\right| \]

      8. fabs-sqr 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right) + 0\right)\right| \]

      9. add-sqr-sqrt 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right) + 0\right)\right| \]

   7. Applied egg-rr 99.9%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right) + 0\right)}\right| \]
   8. Step-by-step derivation

      1. add0 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-plus 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      4. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 67.1%

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

Alternative 3: 99.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 1.0) {
		tmp = x_m * ((2.0 + ((0.2 * Math.pow(x_m, 4.0)) + (0.6666666666666666 * Math.pow(x_m, 2.0)))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 1.0:
		tmp = x_m * ((2.0 + ((0.2 * math.pow(x_m, 4.0)) + (0.6666666666666666 * math.pow(x_m, 2.0)))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))))
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 1.0)
		tmp = x_m * ((2.0 + ((0.2 * (x_m ^ 4.0)) + (0.6666666666666666 * (x_m ^ 2.0)))) / sqrt(pi));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. add0 99.4%

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

      2. add-sqr-sqrt 51.3%

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

      3. fabs-sqr 51.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. add-sqr-sqrt 52.4%

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

      6. fabs-sqr 52.4%

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

      7. add-sqr-sqrt 53.3%

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

      8. +-commutative 53.3%

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

      9. fma-define 53.3%

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

      10. pow2 53.3%

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

   7. Applied egg-rr 53.3%

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

      1. add0 53.3%

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

   9. Simplified 53.3%

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

       1. fma-undefine 53.3%

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

       2. fma-undefine 53.3%

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

       3. associate-+r+ 53.3%

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

   11. Applied egg-rr 53.3%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. add0 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}} + 0\right)}\right| \]

      2. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)} + 0\right)\right| \]

      3. inv-pow 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      4. sqrt-pow1 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      5. metadata-eval 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      6. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)} + 0\right)\right| \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right) + 0\right)\right| \]

      8. fabs-sqr 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right) + 0\right)\right| \]

      9. add-sqr-sqrt 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right) + 0\right)\right| \]

   7. Applied egg-rr 99.9%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right) + 0\right)}\right| \]
   8. Step-by-step derivation

      1. add0 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-plus 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      4. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 67.1%

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

Alternative 4: 99.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 1:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x\_m}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1.0)
   (* (sqrt (/ 1.0 PI)) (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0))))
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1.0) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0)));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 1.0) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((x_m * 2.0) + (0.6666666666666666 * Math.pow(x_m, 3.0)));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 1.0:
		tmp = math.sqrt((1.0 / math.pi)) * ((x_m * 2.0) + (0.6666666666666666 * math.pow(x_m, 3.0)))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1.0)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 1.0)
		tmp = sqrt((1.0 / pi)) * ((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0)));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.0], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. add0 99.4%

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

      2. add-sqr-sqrt 51.3%

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

      3. fabs-sqr 51.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. add-sqr-sqrt 52.4%

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

      6. fabs-sqr 52.4%

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

      7. add-sqr-sqrt 53.3%

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

      8. +-commutative 53.3%

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

      9. fma-define 53.3%

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

      10. pow2 53.3%

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

   7. Applied egg-rr 53.3%

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

      1. add0 53.3%

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

   9. Simplified 53.3%

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

       1. fma-undefine 53.3%

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

       2. fma-undefine 53.3%

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

       3. associate-+r+ 53.3%

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

   11. Applied egg-rr 53.3%

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

   12. Taylor expanded in x around 0 53.1%

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

       1. +-commutative 53.1%

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

       2. associate-*r* 53.1%

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

       3. associate-*r* 53.1%

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

       4. distribute-rgt-out 53.1%

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

   14. Simplified 53.1%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. add0 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}} + 0\right)}\right| \]

      2. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)} + 0\right)\right| \]

      3. inv-pow 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      4. sqrt-pow1 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      5. metadata-eval 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right) + 0\right)\right| \]

      6. *-commutative 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)} + 0\right)\right| \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right) + 0\right)\right| \]

      8. fabs-sqr 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right) + 0\right)\right| \]

      9. add-sqr-sqrt 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right) + 0\right)\right| \]

   7. Applied egg-rr 99.9%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right) + 0\right)}\right| \]
   8. Step-by-step derivation

      1. add0 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-plus 99.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      4. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248}{\pi} \cdot {x\_m}^{14}}\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* (sqrt (/ 1.0 PI)) (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0))))
   (fabs (sqrt (* (/ 0.0022675736961451248 PI) (pow x_m 14.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0)));
	} else {
		tmp = fabs(sqrt(((0.0022675736961451248 / ((double) M_PI)) * pow(x_m, 14.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((x_m * 2.0) + (0.6666666666666666 * Math.pow(x_m, 3.0)));
	} else {
		tmp = Math.abs(Math.sqrt(((0.0022675736961451248 / Math.PI) * Math.pow(x_m, 14.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = math.sqrt((1.0 / math.pi)) * ((x_m * 2.0) + (0.6666666666666666 * math.pow(x_m, 3.0)))
	else:
		tmp = math.fabs(math.sqrt(((0.0022675736961451248 / math.pi) * math.pow(x_m, 14.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))));
	else
		tmp = abs(sqrt(Float64(Float64(0.0022675736961451248 / pi) * (x_m ^ 14.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = sqrt((1.0 / pi)) * ((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0)));
	else
		tmp = abs(sqrt(((0.0022675736961451248 / pi) * (x_m ^ 14.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[Sqrt[N[(N[(0.0022675736961451248 / Pi), $MachinePrecision] * N[Power[x$95$m, 14.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. add0 93.1%

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

      2. add-sqr-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. add-sqr-sqrt 37.5%

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

      5. add-sqr-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. add-sqr-sqrt 37.5%

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

      8. +-commutative 37.5%

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

      9. fma-define 37.5%

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

      10. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. add0 37.5%

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

   9. Simplified 37.5%

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

       1. fma-undefine 37.5%

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

       2. fma-undefine 37.5%

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

       3. associate-+r+ 37.5%

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

   11. Applied egg-rr 37.5%

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

   12. Taylor expanded in x around 0 37.4%

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

       1. +-commutative 37.4%

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

       2. associate-*r* 37.4%

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

       3. associate-*r* 37.4%

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

       4. distribute-rgt-out 37.4%

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

   14. Simplified 37.4%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 33.6%

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

      1. add0 33.6%

         \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0}\right| \]

      2. *-commutative 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} + 0\right| \]

      3. inv-pow 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) + 0\right| \]

      4. sqrt-pow1 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) + 0\right| \]

      5. metadata-eval 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) + 0\right| \]

      6. *-commutative 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right) + 0\right| \]

      7. add-sqr-sqrt 2.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right)\right) + 0\right| \]

      8. fabs-sqr 2.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right)\right) + 0\right| \]

      9. add-sqr-sqrt 33.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right)\right) + 0\right| \]

   7. Applied egg-rr 33.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right) + 0}\right| \]
   8. Step-by-step derivation

      1. associate-*r* 33.6%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {x}^{6}\right)} + 0\right| \]

      2. add0 33.6%

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

   9. Simplified 33.6%

      \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {x}^{6}\right)}\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 3.6%

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

       2. sqrt-unprod 29.9%

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

       3. swap-sqr 29.9%

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

       4. *-commutative 29.9%

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

       5. *-commutative 29.9%

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

       6. swap-sqr 29.9%

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

       7. pow-prod-up 29.9%

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

       8. metadata-eval 29.9%

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

       9. inv-pow 29.9%

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

       10. metadata-eval 29.9%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot \color{blue}{0.0022675736961451248}\right) \cdot \left(\left(x \cdot {x}^{6}\right) \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

       11. pow2 29.9%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot 0.0022675736961451248\right) \cdot \color{blue}{{\left(x \cdot {x}^{6}\right)}^{2}}}\right| \]

       12. pow1 29.9%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot 0.0022675736961451248\right) \cdot {\left(\color{blue}{{x}^{1}} \cdot {x}^{6}\right)}^{2}}\right| \]

       13. pow-prod-up 29.9%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot 0.0022675736961451248\right) \cdot {\color{blue}{\left({x}^{\left(1 + 6\right)}\right)}}^{2}}\right| \]

       14. metadata-eval 29.9%

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

   11. Applied egg-rr 29.9%

       \[\leadsto \left|\color{blue}{\sqrt{\left(\frac{1}{\pi} \cdot 0.0022675736961451248\right) \cdot {\left({x}^{7}\right)}^{2}}}\right| \]
   12. Step-by-step derivation

       1. associate-*l/ 29.9%

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

       2. metadata-eval 29.9%

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

       3. unpow2 29.9%

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

       4. pow-sqr 29.9%

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

       5. metadata-eval 29.9%

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

   13. Simplified 29.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248}{\pi} \cdot {x}^{14}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{0.2 \cdot {x\_m}^{4}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.3)
   (* (sqrt (/ 1.0 PI)) (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0))))
   (* x_m (/ (* 0.2 (pow x_m 4.0)) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0)));
	} else {
		tmp = x_m * ((0.2 * pow(x_m, 4.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((x_m * 2.0) + (0.6666666666666666 * Math.pow(x_m, 3.0)));
	} else {
		tmp = x_m * ((0.2 * Math.pow(x_m, 4.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.3:
		tmp = math.sqrt((1.0 / math.pi)) * ((x_m * 2.0) + (0.6666666666666666 * math.pow(x_m, 3.0)))
	else:
		tmp = x_m * ((0.2 * math.pow(x_m, 4.0)) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.3)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))));
	else
		tmp = Float64(x_m * Float64(Float64(0.2 * (x_m ^ 4.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.3)
		tmp = sqrt((1.0 / pi)) * ((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0)));
	else
		tmp = x_m * ((0.2 * (x_m ^ 4.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.3], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. add0 93.1%

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

      2. add-sqr-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. add-sqr-sqrt 37.5%

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

      5. add-sqr-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. add-sqr-sqrt 37.5%

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

      8. +-commutative 37.5%

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

      9. fma-define 37.5%

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

      10. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. add0 37.5%

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

   9. Simplified 37.5%

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

       1. fma-undefine 37.5%

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

       2. fma-undefine 37.5%

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

       3. associate-+r+ 37.5%

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

   11. Applied egg-rr 37.5%

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

   12. Taylor expanded in x around 0 37.4%

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

       1. +-commutative 37.4%

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

       2. associate-*r* 37.4%

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

       3. associate-*r* 37.4%

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

       4. distribute-rgt-out 37.4%

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

   14. Simplified 37.4%

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

   if 2.2999999999999998 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. add0 93.1%

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

      2. add-sqr-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. add-sqr-sqrt 37.5%

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

      5. add-sqr-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. add-sqr-sqrt 37.5%

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

      8. +-commutative 37.5%

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

      9. fma-define 37.5%

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

      10. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. add0 37.5%

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

   9. Simplified 37.5%

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

   10. Taylor expanded in x around inf 3.8%

       \[\leadsto x \cdot \frac{\color{blue}{0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.78:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x\_m \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{0.2 \cdot {x\_m}^{4}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.78)
   (fabs (* (pow PI -0.5) (* x_m 2.0)))
   (* x_m (/ (* 0.2 (pow x_m 4.0)) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.78) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x_m * 2.0)));
	} else {
		tmp = x_m * ((0.2 * pow(x_m, 4.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.78) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x_m * 2.0)));
	} else {
		tmp = x_m * ((0.2 * Math.pow(x_m, 4.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.78:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x_m * 2.0)))
	else:
		tmp = x_m * ((0.2 * math.pow(x_m, 4.0)) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.78)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x_m * 2.0)));
	else
		tmp = Float64(x_m * Float64(Float64(0.2 * (x_m ^ 4.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.78)
		tmp = abs(((pi ^ -0.5) * (x_m * 2.0)));
	else
		tmp = x_m * ((0.2 * (x_m ^ 4.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.78], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x$95$m * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.78000000000000003

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*l* 71.1%

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

   7. Simplified 71.1%

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

      1. add0 71.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + 0}\right| \]

      2. inv-pow 71.1%

         \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      3. sqrt-pow1 71.1%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      4. metadata-eval 71.1%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      5. add-sqr-sqrt 35.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right) + 0\right| \]

      6. fabs-sqr 35.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right) + 0\right| \]

      7. add-sqr-sqrt 71.1%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot 2\right) + 0\right| \]

   9. Applied egg-rr 71.1%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + 0}\right| \]
   10. Step-by-step derivation

       1. add0 71.1%

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

       2. *-commutative 71.1%

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

   11. Simplified 71.1%

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

   if 1.78000000000000003 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 93.1%

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

      1. add0 93.1%

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

      2. add-sqr-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. add-sqr-sqrt 37.5%

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

      5. add-sqr-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. add-sqr-sqrt 37.5%

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

      8. +-commutative 37.5%

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

      9. fma-define 37.5%

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

      10. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. add0 37.5%

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

   9. Simplified 37.5%

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

   10. Taylor expanded in x around inf 3.8%

       \[\leadsto x \cdot \frac{\color{blue}{0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.78:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x\_m \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \left(x\_m \cdot \frac{x\_m}{\pi}\right)}\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-53)
   (fabs (* (pow PI -0.5) (* x_m 2.0)))
   (fabs (sqrt (* 4.0 (* x_m (/ x_m PI)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-53) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x_m * 2.0)));
	} else {
		tmp = fabs(sqrt((4.0 * (x_m * (x_m / ((double) M_PI))))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 5e-53) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x_m * 2.0)));
	} else {
		tmp = Math.abs(Math.sqrt((4.0 * (x_m * (x_m / Math.PI)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-53:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x_m * 2.0)))
	else:
		tmp = math.fabs(math.sqrt((4.0 * (x_m * (x_m / math.pi)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-53)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x_m * 2.0)));
	else
		tmp = abs(sqrt(Float64(4.0 * Float64(x_m * Float64(x_m / pi)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 5e-53)
		tmp = abs(((pi ^ -0.5) * (x_m * 2.0)));
	else
		tmp = abs(sqrt((4.0 * (x_m * (x_m / pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5e-53], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(4.0 * N[(x$95$m * N[(x$95$m / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e-53

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*l* 69.6%

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

   7. Simplified 69.6%

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

      1. add0 69.6%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + 0}\right| \]

      2. inv-pow 69.6%

         \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      3. sqrt-pow1 69.6%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      4. metadata-eval 69.6%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      5. add-sqr-sqrt 31.9%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right) + 0\right| \]

      6. fabs-sqr 31.9%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right) + 0\right| \]

      7. add-sqr-sqrt 69.6%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot 2\right) + 0\right| \]

   9. Applied egg-rr 69.6%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + 0}\right| \]
   10. Step-by-step derivation

       1. add0 69.6%

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

       2. *-commutative 69.6%

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

   11. Simplified 69.6%

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

   if 5e-53 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 93.9%

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

   7. Simplified 93.9%

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

      1. add0 93.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + 0}\right| \]

      2. inv-pow 93.9%

         \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      3. sqrt-pow1 93.9%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      4. metadata-eval 93.9%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

      5. add-sqr-sqrt 93.4%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right) + 0\right| \]

      6. fabs-sqr 93.4%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right) + 0\right| \]

      7. add-sqr-sqrt 93.9%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot 2\right) + 0\right| \]

   9. Applied egg-rr 93.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + 0}\right| \]
   10. Step-by-step derivation

       1. add0 93.9%

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

       2. *-commutative 93.9%

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

   11. Simplified 93.9%

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

       1. *-commutative 93.9%

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

       2. add-sqr-sqrt 93.1%

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

       3. sqrt-unprod 93.9%

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

       4. swap-sqr 93.5%

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

       5. pow-prod-up 93.9%

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

       6. metadata-eval 93.9%

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

       7. swap-sqr 93.9%

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

       8. unpow2 93.9%

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

       9. metadata-eval 93.9%

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

   13. Applied egg-rr 93.9%

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

       1. *-commutative 93.9%

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

       2. *-commutative 93.9%

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

       3. associate-*l* 93.9%

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

       4. unpow-1 93.9%

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

       5. associate-*r/ 93.9%

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

       6. *-rgt-identity 93.9%

          \[\leadsto \left|\sqrt{4 \cdot \frac{\color{blue}{{x}^{2}}}{\pi}}\right| \]

   15. Simplified 93.9%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}}\right| \]
   16. Step-by-step derivation

       1. unpow2 93.9%

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

       2. associate-/l* 93.9%

          \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(x \cdot \frac{x}{\pi}\right)}}\right| \]

   17. Applied egg-rr 93.9%

       \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(x \cdot \frac{x}{\pi}\right)}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \left(x \cdot \frac{x}{\pi}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.1%

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

   1. *-commutative 71.1%

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

   2. associate-*l* 71.1%

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

7. Simplified 71.1%

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

   1. add0 71.1%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + 0}\right| \]

   2. inv-pow 71.1%

      \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   3. sqrt-pow1 71.1%

      \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   4. metadata-eval 71.1%

      \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   5. add-sqr-sqrt 35.7%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right) + 0\right| \]

   6. fabs-sqr 35.7%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right) + 0\right| \]

   7. add-sqr-sqrt 71.1%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot 2\right) + 0\right| \]

9. Applied egg-rr 71.1%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + 0}\right| \]
10. Step-by-step derivation

    1. add0 71.1%

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

    2. *-commutative 71.1%

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

11. Simplified 71.1%

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

12. Final simplification 71.1%

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

Alternative 10: 67.8% accurate, 9.0× speedup

Math

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

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(Float64(x_m * 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[(N[(x$95$m * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.1%

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

   1. *-commutative 71.1%

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

   2. associate-*l* 71.1%

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

7. Simplified 71.1%

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

   1. add0 71.1%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + 0}\right| \]

   2. inv-pow 71.1%

      \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   3. sqrt-pow1 71.1%

      \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   4. metadata-eval 71.1%

      \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot 2\right) + 0\right| \]

   5. add-sqr-sqrt 35.7%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right) + 0\right| \]

   6. fabs-sqr 35.7%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right) + 0\right| \]

   7. add-sqr-sqrt 71.1%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot 2\right) + 0\right| \]

9. Applied egg-rr 71.1%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + 0}\right| \]
10. Step-by-step derivation

    1. add0 71.1%

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

    2. *-commutative 71.1%

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

11. Simplified 71.1%

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

    1. *-commutative 71.1%

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

    2. add-sqr-sqrt 35.6%

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

    3. sqrt-unprod 54.1%

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

    4. swap-sqr 54.0%

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

    5. pow-prod-up 54.1%

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

    6. metadata-eval 54.1%

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

    7. swap-sqr 54.1%

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

    8. unpow2 54.1%

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

    9. metadata-eval 54.1%

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

13. Applied egg-rr 54.1%

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

    1. *-commutative 54.1%

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

    2. *-commutative 54.1%

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

    3. associate-*l* 54.0%

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

    4. unpow-1 54.0%

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

    5. associate-*r/ 54.1%

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

    6. *-rgt-identity 54.1%

       \[\leadsto \left|\sqrt{4 \cdot \frac{\color{blue}{{x}^{2}}}{\pi}}\right| \]

15. Simplified 54.1%

    \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}}\right| \]
16. Step-by-step derivation

    1. add0 54.1%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \frac{{x}^{2}}{\pi}} + 0}\right| \]

    2. pow1 54.1%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}\right)}^{1}} + 0\right| \]

    3. pow1 54.1%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}} + 0\right| \]

    4. sqrt-prod 54.1%

       \[\leadsto \left|\color{blue}{\sqrt{4} \cdot \sqrt{\frac{{x}^{2}}{\pi}}} + 0\right| \]

    5. metadata-eval 54.1%

       \[\leadsto \left|\color{blue}{2} \cdot \sqrt{\frac{{x}^{2}}{\pi}} + 0\right| \]

    6. sqrt-div 53.9%

       \[\leadsto \left|2 \cdot \color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{\pi}}} + 0\right| \]

    7. unpow2 53.9%

       \[\leadsto \left|2 \cdot \frac{\sqrt{\color{blue}{x \cdot x}}}{\sqrt{\pi}} + 0\right| \]

    8. sqrt-prod 35.6%

       \[\leadsto \left|2 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}} + 0\right| \]

    9. add-sqr-sqrt 70.7%

       \[\leadsto \left|2 \cdot \frac{\color{blue}{x}}{\sqrt{\pi}} + 0\right| \]

17. Applied egg-rr 70.7%

    \[\leadsto \left|\color{blue}{2 \cdot \frac{x}{\sqrt{\pi}} + 0}\right| \]
18. Step-by-step derivation

    1. add0 70.7%

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

    2. *-commutative 70.7%

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

    3. associate-*l/ 70.7%

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

19. Simplified 70.7%

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

20. Final simplification 70.7%

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

Alternative 11: 4.2% accurate, 1849.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.1%

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

   1. *-commutative 71.1%

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

   2. associate-*l* 71.1%

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

7. Simplified 71.1%

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

   1. expm1-log1p-u 71.1%

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

   2. expm1-undefine 6.9%

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

   3. inv-pow 6.9%

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

   4. sqrt-pow1 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-sqr-sqrt 2.6%

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

   7. fabs-sqr 2.6%

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

   8. add-sqr-sqrt 5.2%

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

9. Applied egg-rr 5.2%

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

10. Taylor expanded in x around 0 4.2%

    \[\leadsto \left|\color{blue}{1} - 1\right| \]

11. Final simplification 4.2%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))))