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

Percentage Accurate: 99.8% → 99.9%

Time: 11.2s

Alternatives: 12

Speedup: 2.4×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. distribute-lft-in 99.5%

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

5. Applied egg-rr 99.5%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.2 \cdot {x}^{5}\\ t_1 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|t_1 \cdot \left(\left(x + x\right) + \left(0.6666666666666666 \cdot {x}^{3} + t_0\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \left(t_0 + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.2 (pow x 5.0))) (t_1 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.5)
     (fabs (* t_1 (+ (+ x x) (+ (* 0.6666666666666666 (pow x 3.0)) t_0))))
     (fabs (* t_1 (+ t_0 (* 0.047619047619047616 (pow x 7.0))))))))

C

double code(double x) {
	double t_0 = 0.2 * pow(x, 5.0);
	double t_1 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs((t_1 * ((x + x) + ((0.6666666666666666 * pow(x, 3.0)) + t_0))));
	} else {
		tmp = fabs((t_1 * (t_0 + (0.047619047619047616 * pow(x, 7.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.2 * Math.pow(x, 5.0);
	double t_1 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 0.5) {
		tmp = Math.abs((t_1 * ((x + x) + ((0.6666666666666666 * Math.pow(x, 3.0)) + t_0))));
	} else {
		tmp = Math.abs((t_1 * (t_0 + (0.047619047619047616 * Math.pow(x, 7.0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.2 * math.pow(x, 5.0)
	t_1 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 0.5:
		tmp = math.fabs((t_1 * ((x + x) + ((0.6666666666666666 * math.pow(x, 3.0)) + t_0))))
	else:
		tmp = math.fabs((t_1 * (t_0 + (0.047619047619047616 * math.pow(x, 7.0)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.2 * (x ^ 5.0))
	t_1 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(t_1 * Float64(Float64(x + x) + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + t_0))));
	else
		tmp = abs(Float64(t_1 * Float64(t_0 + Float64(0.047619047619047616 * (x ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.2 * (x ^ 5.0);
	t_1 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 0.5)
		tmp = abs((t_1 * ((x + x) + ((0.6666666666666666 * (x ^ 3.0)) + t_0))));
	else
		tmp = abs((t_1 * (t_0 + (0.047619047619047616 * (x ^ 7.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(t$95$1 * N[(N[(x + x), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[(t$95$0 + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   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 99.3%

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

      1. distribute-lft-in 99.3%

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

   5. Applied egg-rr 99.3%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. associate-+r+ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*l* 99.5%

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

      6. associate-*r* 99.5%

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

      7. associate-*r* 99.5%

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

      8. distribute-rgt-out 99.5%

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

      9. distribute-lft-out 99.5%

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

   10. Simplified 99.5%

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

       1. fma-udef 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 99.9%

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

   5. Applied egg-rr 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.2%

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

      1. associate-*r* 99.2%

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

      2. associate-*r* 99.2%

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

      3. distribute-rgt-out 99.2%

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

   10. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(x + x\right) + \left(0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. div-inv 99.2%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 98.9%

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

   2. *-rgt-identity 98.9%

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

   3. unpow1 98.9%

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

   4. sqr-pow 29.9%

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

   5. fabs-sqr 29.9%

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

   6. sqr-pow 98.9%

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

   7. unpow1 98.9%

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

7. Simplified 99.5%

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

   1. fma-udef 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 4: 99.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (sqrt (/ 1.0 PI)))
   (+ 2.0 (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))))))

C

double code(double x) {
	return fabs(((x * sqrt((1.0 / ((double) M_PI)))) * (2.0 + fma(0.2, pow(x, 4.0), (0.047619047619047616 * pow(x, 6.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(x * sqrt(Float64(1.0 / pi))) * Float64(2.0 + fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in x around 0 99.2%

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

   1. unpow1 99.2%

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

   2. sqr-pow 30.0%

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

   3. fabs-sqr 30.0%

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

   4. sqr-pow 99.2%

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

   5. unpow1 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 98.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 98.9%

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

   1. div-inv 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-*r/ 98.9%

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

   2. *-rgt-identity 98.9%

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

   3. unpow1 98.9%

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

   4. sqr-pow 29.9%

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

   5. fabs-sqr 29.9%

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

   6. sqr-pow 98.9%

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

   7. unpow1 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 89.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.85)
     (fabs (* t_0 (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
     (fabs
      (* t_0 (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((t_0 * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = fabs((t_0 * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((t_0 * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = Math.abs((t_0 * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((t_0 * (x * (2.0 + (x * (x * 0.6666666666666666))))))
	else:
		tmp = math.fabs((t_0 * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(t_0 * Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = abs(Float64(t_0 * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((t_0 * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	else
		tmp = abs((t_0 * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.85], N[Abs[N[(t$95$0 * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. expm1-log1p-u 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-udef 31.8%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      3. +-commutative 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      4. fma-def 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

   6. Applied egg-rr 31.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-log1p 83.6%

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

      3. fma-udef 83.6%

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

      4. unpow3 83.6%

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

      5. sqr-abs 83.6%

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

      6. unpow2 83.6%

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

      7. associate-*r* 83.6%

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

      8. unpow2 83.6%

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

      9. associate-*r* 83.6%

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

      10. *-commutative 83.6%

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

      11. distribute-rgt-out 83.6%

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

      12. unpow1 83.6%

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

      13. sqr-pow 30.0%

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

      14. fabs-sqr 30.0%

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

      15. sqr-pow 83.6%

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

      16. unpow1 83.6%

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

      17. fma-def 83.6%

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

   8. Simplified 83.6%

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

      1. fma-udef 83.6%

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

   10. Applied egg-rr 83.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

   5. Applied egg-rr 99.5%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 45.6%

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

      1. associate-*r* 45.6%

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

      2. associate-*r* 45.7%

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

      3. distribute-rgt-out 45.7%

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

   10. Simplified 45.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 7: 89.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (fabs x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * fabs(x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = Math.abs((0.047619047619047616 * ((Math.pow(x, 6.0) * Math.abs(x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))))
	else:
		tmp = math.fabs((0.047619047619047616 * ((math.pow(x, 6.0) * math.fabs(x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	else
		tmp = abs((0.047619047619047616 * (((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. expm1-log1p-u 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-udef 31.8%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      3. +-commutative 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      4. fma-def 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

   6. Applied egg-rr 31.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-log1p 83.6%

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

      3. fma-udef 83.6%

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

      4. unpow3 83.6%

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

      5. sqr-abs 83.6%

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

      6. unpow2 83.6%

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

      7. associate-*r* 83.6%

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

      8. unpow2 83.6%

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

      9. associate-*r* 83.6%

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

      10. *-commutative 83.6%

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

      11. distribute-rgt-out 83.6%

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

      12. unpow1 83.6%

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

      13. sqr-pow 30.0%

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

      14. fabs-sqr 30.0%

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

      15. sqr-pow 83.6%

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

      16. unpow1 83.6%

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

      17. fma-def 83.6%

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

   8. Simplified 83.6%

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

      1. fma-udef 83.6%

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

   10. Applied egg-rr 83.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 45.4%

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

      1. expm1-log1p-u 44.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      2. expm1-udef 44.6%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

      3. *-commutative 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)}\right)} - 1\right| \]

      4. sqrt-div 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      5. metadata-eval 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      6. *-commutative 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left({x}^{6} \cdot \left|x\right|\right)}\right)\right)} - 1\right| \]

   6. Applied egg-rr 44.6%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 44.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right)\right)}\right| \]

      2. expm1-log1p 45.4%

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

      3. associate-*l/ 45.4%

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

      4. *-lft-identity 45.4%

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

      5. *-commutative 45.4%

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

   8. Simplified 45.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8: 89.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. expm1-log1p-u 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-udef 31.8%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      3. +-commutative 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      4. fma-def 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

   6. Applied egg-rr 31.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-log1p 83.6%

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

      3. fma-udef 83.6%

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

      4. unpow3 83.6%

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

      5. sqr-abs 83.6%

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

      6. unpow2 83.6%

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

      7. associate-*r* 83.6%

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

      8. unpow2 83.6%

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

      9. associate-*r* 83.6%

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

      10. *-commutative 83.6%

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

      11. distribute-rgt-out 83.6%

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

      12. unpow1 83.6%

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

      13. sqr-pow 30.0%

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

      14. fabs-sqr 30.0%

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

      15. sqr-pow 83.6%

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

      16. unpow1 83.6%

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

      17. fma-def 83.6%

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

   8. Simplified 83.6%

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

      1. fma-udef 83.6%

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

   10. Applied egg-rr 83.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 45.4%

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

      1. expm1-log1p-u 44.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      2. expm1-udef 44.6%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

      3. *-commutative 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)}\right)} - 1\right| \]

      4. sqrt-div 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      5. metadata-eval 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      6. *-commutative 44.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left({x}^{6} \cdot \left|x\right|\right)}\right)\right)} - 1\right| \]

   6. Applied egg-rr 44.6%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 44.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right)\right)}\right| \]

      2. expm1-log1p 45.4%

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

      3. associate-*l/ 45.4%

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

      4. *-lft-identity 45.4%

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

      5. unpow1 45.4%

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

      6. sqr-pow 1.8%

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

      7. fabs-sqr 1.8%

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

      8. sqr-pow 45.4%

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

      9. unpow1 45.4%

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

      10. pow-plus 45.4%

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

      11. metadata-eval 45.4%

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

      12. unpow1 45.4%

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

      13. sqr-pow 1.8%

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

      14. fabs-sqr 1.8%

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

      15. sqr-pow 45.4%

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

      16. unpow1 45.4%

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

   8. Simplified 45.4%

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

4. Final simplification 83.6%

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

Alternative 9: 68.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.72)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (sqrt (/ 1.0 PI)) (* x (* 0.6666666666666666 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.72) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (0.6666666666666666 * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.72) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (0.6666666666666666 * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.72:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (0.6666666666666666 * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.72)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(0.6666666666666666 * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.72)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((sqrt((1.0 / pi)) * (x * (0.6666666666666666 * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.72], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.71999999999999997

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

      3. unpow1 59.4%

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

      4. sqr-pow 30.0%

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

      5. fabs-sqr 30.0%

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

      6. sqr-pow 59.4%

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

      7. unpow1 59.4%

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

   6. Simplified 59.4%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 5.2%

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

      3. *-commutative 5.2%

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

      4. sqrt-div 5.2%

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

      5. metadata-eval 5.2%

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

   8. Applied egg-rr 5.2%

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

      1. expm1-def 57.0%

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

      2. expm1-log1p 59.4%

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

      3. associate-*r/ 59.4%

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

      4. metadata-eval 59.4%

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

   10. Simplified 59.4%

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

   if 1.71999999999999997 < x

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. expm1-log1p-u 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-udef 31.8%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      3. +-commutative 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      4. fma-def 31.8%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

   6. Applied egg-rr 31.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      2. expm1-log1p 83.6%

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

      3. fma-udef 83.6%

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

      4. unpow3 83.6%

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

      5. sqr-abs 83.6%

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

      6. unpow2 83.6%

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

      7. associate-*r* 83.6%

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

      8. unpow2 83.6%

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

      9. associate-*r* 83.6%

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

      10. *-commutative 83.6%

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

      11. distribute-rgt-out 83.6%

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

      12. unpow1 83.6%

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

      13. sqr-pow 30.0%

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

      14. fabs-sqr 30.0%

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

      15. sqr-pow 83.6%

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

      16. unpow1 83.6%

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

      17. fma-def 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in x around inf 30.2%

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

       1. unpow2 30.2%

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

   11. Simplified 30.2%

       \[\leadsto \left|\left(x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

Alternative 10: 89.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 83.6%

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

   1. expm1-log1p-u 83.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

   2. expm1-udef 31.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

   3. +-commutative 31.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

   4. fma-def 31.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)} - 1\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

6. Applied egg-rr 31.8%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)} - 1\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]
7. Step-by-step derivation

   1. expm1-def 83.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

   2. expm1-log1p 83.6%

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

   3. fma-udef 83.6%

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

   4. unpow3 83.6%

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

   5. sqr-abs 83.6%

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

   6. unpow2 83.6%

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

   7. associate-*r* 83.6%

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

   8. unpow2 83.6%

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

   9. associate-*r* 83.6%

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

   10. *-commutative 83.6%

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

   11. distribute-rgt-out 83.6%

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

   12. unpow1 83.6%

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

   13. sqr-pow 30.0%

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

   14. fabs-sqr 30.0%

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

   15. sqr-pow 83.6%

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

   16. unpow1 83.6%

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

   17. fma-def 83.6%

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

8. Simplified 83.6%

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

   1. fma-udef 83.6%

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

10. Applied egg-rr 83.6%

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

11. Final simplification 83.6%

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

Alternative 11: 68.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(x + x\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (sqrt (/ 1.0 PI)) (+ x x))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (x + x)));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (x + x)));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x + x)))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x + x)))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (x + x)));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. distribute-lft-in 99.5%

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

5. Applied egg-rr 99.5%

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

7. Simplified 99.9%

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

8. Taylor expanded in x around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. *-commutative 59.4%

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

   3. unpow1 59.4%

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

   4. sqr-pow 30.0%

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

   5. fabs-sqr 30.0%

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

   6. sqr-pow 59.4%

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

   7. unpow1 59.4%

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

   8. associate-*l* 59.7%

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

   9. *-commutative 59.7%

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

   10. rem-log-exp 40.0%

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

   11. *-commutative 40.0%

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

   12. exp-lft-sqr 39.9%

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

   13. log-prod 39.9%

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

   14. unpow1 39.9%

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

   15. sqr-pow 2.1%

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

   16. fabs-sqr 2.1%

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

   17. sqr-pow 3.8%

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

   18. unpow1 3.8%

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

   19. rem-log-exp 45.5%

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

   20. unpow1 45.5%

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

10. Simplified 59.7%

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

11. Final simplification 59.7%

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

Alternative 12: 68.5% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. associate-*l* 59.4%

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

   3. unpow1 59.4%

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

   4. sqr-pow 30.0%

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

   5. fabs-sqr 30.0%

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

   6. sqr-pow 59.4%

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

   7. unpow1 59.4%

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

6. Simplified 59.4%

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

   1. expm1-log1p-u 57.0%

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

   2. expm1-udef 5.2%

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

   3. *-commutative 5.2%

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

   4. sqrt-div 5.2%

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

   5. metadata-eval 5.2%

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

8. Applied egg-rr 5.2%

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

   1. expm1-def 57.0%

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

   2. expm1-log1p 59.4%

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

   3. associate-*r/ 59.4%

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

   4. metadata-eval 59.4%

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

10. Simplified 59.4%

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

11. Final simplification 59.4%

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

Reproduce

herbie shell --seed 2023275 
(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)))))))