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

Percentage Accurate: 99.9% → 99.9%

Time: 11.0s

Alternatives: 11

Speedup: 3.0×

• 28.8% 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: 99.9% 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, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 31.0%

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

   2. fabs-sqr 31.0%

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

   3. add-sqr-sqrt 32.5%

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

   4. *-un-lft-identity 32.5%

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

6. Applied egg-rr 32.5%

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

   1. *-lft-identity 32.5%

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

8. Simplified 32.5%

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

Alternative 2: 99.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left|x\_m\right|\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right) + 0.2 \cdot t\_0\right) + 0.047619047619047616 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot t\_0\right)\right)\right| \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) (* (* x_m x_m) (fabs x_m)))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0))) (* 0.2 t_0))
      (* 0.047619047619047616 (* (* x_m x_m) t_0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (x_m * x_m) * ((x_m * x_m) * fabs(x_m));
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x_m * x_m) * t_0)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (x_m * x_m) * ((x_m * x_m) * Math.abs(x_m));
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((x_m * 2.0) + (0.6666666666666666 * Math.pow(x_m, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x_m * x_m) * t_0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (x_m * x_m) * ((x_m * x_m) * math.fabs(x_m))
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((x_m * 2.0) + (0.6666666666666666 * math.pow(x_m, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x_m * x_m) * t_0)))))

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * abs(x_m)))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))) + Float64(0.2 * t_0)) + Float64(0.047619047619047616 * Float64(Float64(x_m * x_m) * t_0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = (x_m * x_m) * ((x_m * x_m) * abs(x_m));
	tmp = abs(((1.0 / sqrt(pi)) * ((((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x_m * x_m) * t_0)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 31.0%

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

   3. fabs-sqr 31.0%

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

   4. add-sqr-sqrt 99.7%

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

   5. associate-*r* 99.7%

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

   6. add-sqr-sqrt 31.2%

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

   7. fabs-sqr 31.2%

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

   8. add-sqr-sqrt 74.6%

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

   9. associate-*r* 74.6%

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

   10. cube-mult 74.6%

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

6. Applied egg-rr 74.6%

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

7. Final simplification 74.6%

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

Alternative 3: 99.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(((2.0 + (pow(x_m, 4.0) * fma((x_m * x_m), 0.047619047619047616, 0.2))) * (x_m * sqrt((1.0 / ((double) M_PI))))));
}

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(Float64(2.0 + Float64((x_m ^ 4.0) * fma(Float64(x_m * x_m), 0.047619047619047616, 0.2))) * Float64(x_m * sqrt(Float64(1.0 / pi)))))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-square-sqrt 30.9%

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

   4. fabs-sqr 30.9%

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

   5. rem-square-sqrt 99.5%

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

   6. associate-*l* 99.5%

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

7. Simplified 99.5%

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

   1. unpow2 99.5%

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

9. Applied egg-rr 99.5%

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

Alternative 4: 98.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.abs(((x_m * Math.sqrt((1.0 / Math.PI))) * ((0.047619047619047616 * Math.pow(x_m, 6.0)) + 2.0)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(((x_m * math.sqrt((1.0 / math.pi))) * ((0.047619047619047616 * math.pow(x_m, 6.0)) + 2.0)))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(((x_m * sqrt((1.0 / pi))) * ((0.047619047619047616 * (x_m ^ 6.0)) + 2.0)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-square-sqrt 30.9%

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

   4. fabs-sqr 30.9%

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

   5. rem-square-sqrt 99.5%

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

   6. associate-*l* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around inf 99.4%

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

9. Final simplification 99.4%

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

Alternative 5: 98.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-square-sqrt 30.9%

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

   4. fabs-sqr 30.9%

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

   5. rem-square-sqrt 99.5%

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

   6. associate-*l* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around inf 99.4%

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

   1. sqrt-div 99.4%

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

   2. metadata-eval 99.4%

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

   3. un-div-inv 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 6: 98.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x_m, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow-1 64.8%

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

      4. metadata-eval 64.8%

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

      5. pow-sqr 64.8%

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

      6. rem-sqrt-square 64.8%

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

      7. rem-square-sqrt 64.8%

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

      8. fabs-sqr 64.8%

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

      9. rem-square-sqrt 64.8%

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

      10. associate-*r* 64.8%

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

   10. Simplified 64.8%

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

       1. *-un-lft-identity 64.8%

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

       2. add-sqr-sqrt 30.9%

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

       3. fabs-sqr 30.9%

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

       4. add-sqr-sqrt 32.5%

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

       5. *-commutative 32.5%

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

       6. metadata-eval 32.5%

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

       7. sqrt-pow1 32.5%

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

       8. inv-pow 32.5%

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

       9. associate-*r* 32.5%

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

       10. sqrt-div 32.5%

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

       11. metadata-eval 32.5%

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

       12. un-div-inv 32.2%

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

   12. Applied egg-rr 32.2%

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

       1. *-lft-identity 32.2%

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

       2. associate-*l/ 32.2%

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

       3. associate-/l* 32.5%

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

   14. Simplified 32.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 40.2%

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

      1. add-sqr-sqrt 3.3%

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

      2. fabs-sqr 3.3%

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

      3. add-sqr-sqrt 3.5%

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

      4. associate-*r* 3.5%

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

      5. sqrt-div 3.5%

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

      6. metadata-eval 3.5%

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

      7. un-div-inv 3.5%

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

   10. Applied egg-rr 3.5%

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

Alternative 7: 98.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow-1 64.8%

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

      4. metadata-eval 64.8%

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

      5. pow-sqr 64.8%

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

      6. rem-sqrt-square 64.8%

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

      7. rem-square-sqrt 64.8%

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

      8. fabs-sqr 64.8%

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

      9. rem-square-sqrt 64.8%

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

      10. associate-*r* 64.8%

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

   10. Simplified 64.8%

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

       1. *-un-lft-identity 64.8%

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

       2. add-sqr-sqrt 30.9%

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

       3. fabs-sqr 30.9%

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

       4. add-sqr-sqrt 32.5%

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

       5. *-commutative 32.5%

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

       6. metadata-eval 32.5%

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

       7. sqrt-pow1 32.5%

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

       8. inv-pow 32.5%

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

       9. associate-*r* 32.5%

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

       10. sqrt-div 32.5%

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

       11. metadata-eval 32.5%

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

       12. un-div-inv 32.2%

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

   12. Applied egg-rr 32.2%

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

       1. *-lft-identity 32.2%

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

       2. associate-*l/ 32.2%

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

       3. associate-/l* 32.5%

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

   14. Simplified 32.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 40.2%

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

      1. add-sqr-sqrt 3.3%

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

      2. fabs-sqr 3.3%

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

      3. add-sqr-sqrt 3.5%

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

      4. sqrt-div 3.5%

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

      5. metadata-eval 3.5%

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

      6. un-div-inv 3.5%

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

   10. Applied egg-rr 3.5%

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

       1. *-commutative 3.5%

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

       2. associate-*l/ 3.5%

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

       3. associate-/l* 3.5%

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

   12. Simplified 3.5%

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

Alternative 8: 98.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow-1 64.8%

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

      4. metadata-eval 64.8%

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

      5. pow-sqr 64.8%

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

      6. rem-sqrt-square 64.8%

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

      7. rem-square-sqrt 64.8%

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

      8. fabs-sqr 64.8%

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

      9. rem-square-sqrt 64.8%

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

      10. associate-*r* 64.8%

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

   10. Simplified 64.8%

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

       1. *-un-lft-identity 64.8%

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

       2. add-sqr-sqrt 30.9%

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

       3. fabs-sqr 30.9%

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

       4. add-sqr-sqrt 32.5%

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

       5. *-commutative 32.5%

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

       6. metadata-eval 32.5%

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

       7. sqrt-pow1 32.5%

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

       8. inv-pow 32.5%

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

       9. associate-*r* 32.5%

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

       10. sqrt-div 32.5%

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

       11. metadata-eval 32.5%

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

       12. un-div-inv 32.2%

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

   12. Applied egg-rr 32.2%

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

       1. *-lft-identity 32.2%

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

       2. associate-*l/ 32.2%

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

       3. associate-/l* 32.5%

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

   14. Simplified 32.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 40.2%

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

      1. add-sqr-sqrt 3.3%

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

      2. fabs-sqr 3.3%

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

      3. add-sqr-sqrt 3.5%

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

      4. *-commutative 3.5%

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

      5. sqrt-div 3.5%

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

      6. metadata-eval 3.5%

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

      7. un-div-inv 3.5%

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

   10. Applied egg-rr 3.5%

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

4. Final simplification 32.5%

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

Alternative 9: 94.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.75:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.4444444444444444 \cdot \frac{{x\_m}^{6}}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.75)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (* 0.4444444444444444 (/ (pow x_m 6.0) PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.4444444444444444 * (pow(x_m, 6.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.4444444444444444 * (Math.pow(x_m, 6.0) / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.75:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.4444444444444444 * (math.pow(x_m, 6.0) / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.75)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.4444444444444444 * Float64((x_m ^ 6.0) / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.75)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.4444444444444444 * ((x_m ^ 6.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.75], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.4444444444444444 * N[(N[Power[x$95$m, 6.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. rem-square-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. rem-square-sqrt 99.5%

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

      6. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow-1 64.8%

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

      4. metadata-eval 64.8%

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

      5. pow-sqr 64.8%

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

      6. rem-sqrt-square 64.8%

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

      7. rem-square-sqrt 64.8%

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

      8. fabs-sqr 64.8%

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

      9. rem-square-sqrt 64.8%

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

      10. associate-*r* 64.8%

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

   10. Simplified 64.8%

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

       1. *-un-lft-identity 64.8%

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

       2. add-sqr-sqrt 30.9%

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

       3. fabs-sqr 30.9%

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

       4. add-sqr-sqrt 32.5%

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

       5. *-commutative 32.5%

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

       6. metadata-eval 32.5%

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

       7. sqrt-pow1 32.5%

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

       8. inv-pow 32.5%

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

       9. associate-*r* 32.5%

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

       10. sqrt-div 32.5%

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

       11. metadata-eval 32.5%

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

       12. un-div-inv 32.2%

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

   12. Applied egg-rr 32.2%

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

       1. *-lft-identity 32.2%

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

       2. associate-*l/ 32.2%

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

       3. associate-/l* 32.5%

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

   14. Simplified 32.5%

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

   if 1.75 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 29.6%

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

      1. *-commutative 29.6%

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

      2. *-commutative 29.6%

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

      3. associate-*l* 29.6%

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

      4. rem-square-sqrt 1.9%

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

      5. fabs-sqr 1.9%

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

      6. rem-square-sqrt 29.6%

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

      7. pow-plus 29.6%

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

      8. metadata-eval 29.6%

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

      9. *-commutative 29.6%

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

   7. Simplified 29.6%

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

      1. add-sqr-sqrt 3.3%

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

      2. fabs-sqr 3.3%

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

      3. sqrt-unprod 34.7%

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

      4. swap-sqr 34.7%

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

      5. add-sqr-sqrt 34.7%

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

      6. *-commutative 34.7%

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

      7. *-commutative 34.7%

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

      8. swap-sqr 34.7%

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

      9. pow-sqr 34.7%

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

      10. metadata-eval 34.7%

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

      11. metadata-eval 34.7%

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

   9. Applied egg-rr 34.7%

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

       1. associate-*r* 34.7%

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

       2. *-commutative 34.7%

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

       3. associate-*l/ 34.7%

          \[\leadsto \sqrt{0.4444444444444444 \cdot \color{blue}{\frac{1 \cdot {x}^{6}}{\pi}}} \]

       4. *-lft-identity 34.7%

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

   11. Simplified 34.7%

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

Alternative 10: 67.8% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-square-sqrt 30.9%

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

   4. fabs-sqr 30.9%

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

   5. rem-square-sqrt 99.5%

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

   6. associate-*l* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 64.8%

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

   1. associate-*r* 64.8%

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

   2. *-commutative 64.8%

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

   3. unpow-1 64.8%

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

   4. metadata-eval 64.8%

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

   5. pow-sqr 64.8%

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

   6. rem-sqrt-square 64.8%

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

   7. rem-square-sqrt 64.8%

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

   8. fabs-sqr 64.8%

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

   9. rem-square-sqrt 64.8%

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

   10. associate-*r* 64.8%

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

10. Simplified 64.8%

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

    1. *-un-lft-identity 64.8%

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

    2. add-sqr-sqrt 30.9%

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

    3. fabs-sqr 30.9%

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

    4. add-sqr-sqrt 32.5%

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

    5. *-commutative 32.5%

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

    6. metadata-eval 32.5%

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

    7. sqrt-pow1 32.5%

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

    8. inv-pow 32.5%

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

    9. associate-*r* 32.5%

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

    10. sqrt-div 32.5%

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

    11. metadata-eval 32.5%

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

    12. un-div-inv 32.2%

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

12. Applied egg-rr 32.2%

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

    1. *-lft-identity 32.2%

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

    2. associate-*l/ 32.2%

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

    3. associate-/l* 32.5%

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

14. Simplified 32.5%

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

Alternative 11: 4.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{expm1}\left(0\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (expm1 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	return expm1(0.0);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.expm1(0.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.expm1(0.0)

Julia

x_m = abs(x)
function code(x_m)
	return expm1(0.0)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(Exp[0.0] - 1), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-square-sqrt 30.9%

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

   4. fabs-sqr 30.9%

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

   5. rem-square-sqrt 99.5%

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

   6. associate-*l* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around inf 40.2%

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

   1. expm1-log1p-u 39.9%

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

   2. add-sqr-sqrt 3.3%

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

   3. fabs-sqr 3.3%

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

   4. add-sqr-sqrt 3.5%

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

   5. associate-*r* 3.5%

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

   6. sqrt-div 3.5%

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

   7. metadata-eval 3.5%

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

   8. un-div-inv 3.5%

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

10. Applied egg-rr 3.5%

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

11. Taylor expanded in x around 0 4.0%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{0}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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)))))))