Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 15.3s

Alternatives: 13

Speedup: 4.2×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} + -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (+
    (exp (log1p (/ (pow x -4.0) (/ x (fma 1.875 (pow x -2.0) 0.75)))))
    -1.0))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), (exp(log1p((pow(x, -4.0) / (x / fma(1.875, pow(x, -2.0), 0.75))))) + -1.0));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(exp(log1p(Float64((x ^ -4.0) / Float64(x / fma(1.875, (x ^ -2.0), 0.75))))) + -1.0)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(N[Power[x, -4.0], $MachinePrecision] / N[(x / N[(1.875 * N[Power[x, -2.0], $MachinePrecision] + 0.75), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   3. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{4}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   4. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   6. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({x}^{\color{blue}{-4}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({x}^{-4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-log1p 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{{x}^{-4}}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{{x}^{-4}}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)\right)}\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} - 1}\right) \]

10. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} + -1\right) \]

Alternative 2: 100.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(1 + \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right) + -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (+ (+ 1.0 (* (fma 1.875 (pow x -2.0) 0.75) (pow x -5.0))) -1.0))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), ((1.0 + (fma(1.875, pow(x, -2.0), 0.75) * pow(x, -5.0))) + -1.0));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(Float64(1.0 + Float64(fma(1.875, (x ^ -2.0), 0.75) * (x ^ -5.0))) + -1.0)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(1.875 * N[Power[x, -2.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   3. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{4}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   4. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   6. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({x}^{\color{blue}{-4}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({x}^{-4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-log1p 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{{x}^{-4}}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{{x}^{-4}}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)\right)}\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} - 1}\right) \]
10. Step-by-step derivation

    1. sub-neg 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} + \left(-1\right)}\right) \]

    2. log1p-udef 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\color{blue}{\log \left(1 + \frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)}} + \left(-1\right)\right) \]

    3. add-exp-log 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\left(1 + \frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}}\right)} + \left(-1\right)\right) \]

    4. +-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\left(\frac{{x}^{-4}}{\frac{x}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}} + 1\right)} + \left(-1\right)\right) \]

    5. associate-/r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\color{blue}{\frac{{x}^{-4}}{x} \cdot \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)} + 1\right) + \left(-1\right)\right) \]

    6. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\color{blue}{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot \frac{{x}^{-4}}{x}} + 1\right) + \left(-1\right)\right) \]

    7. pow1 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot \frac{{x}^{-4}}{\color{blue}{{x}^{1}}} + 1\right) + \left(-1\right)\right) \]

    8. pow-div 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot \color{blue}{{x}^{\left(-4 - 1\right)}} + 1\right) + \left(-1\right)\right) \]

    9. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{\color{blue}{-5}} + 1\right) + \left(-1\right)\right) \]

    10. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5} + 1\right) + \color{blue}{-1}\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5} + 1\right) + -1}\right) \]

12. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \left(1 + \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right) + -1\right) \]

Alternative 3: 100.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (/ (pow (exp x) x) (fabs x)) (sqrt PI))
  (+ 1.0 (+ (* 1.875 (pow x -6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))))

C

double code(double x) {
	return ((pow(exp(x), x) / fabs(x)) / sqrt(((double) M_PI))) * (1.0 + ((1.875 * pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Java

public static double code(double x) {
	return ((Math.pow(Math.exp(x), x) / Math.abs(x)) / Math.sqrt(Math.PI)) * (1.0 + ((1.875 * Math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Python

def code(x):
	return ((math.pow(math.exp(x), x) / math.fabs(x)) / math.sqrt(math.pi)) * (1.0 + ((1.875 * math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))))

Julia

function code(x)
	return Float64(Float64(Float64((exp(x) ^ x) / abs(x)) / sqrt(pi)) * Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = (((exp(x) ^ x) / abs(x)) / sqrt(pi)) * (1.0 + ((1.875 * (x ^ -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
end

Wolfram

code[x_] := N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{x}^{6}}{1.875}}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. associate-/r/ 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{x}^{6}} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. pow-flip 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{\left(-6\right)}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-6}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-6} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

6. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{\frac{\color{blue}{e^{{x}^{2}}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. exp-prod 100.0%

      \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

9. Final simplification 100.0%

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

Alternative 4: 100.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (* 1.875 (pow x -6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))
  (/ (/ (exp (* x x)) (fabs x)) (sqrt PI))))

C

double code(double x) {
	return (1.0 + ((1.875 * pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * ((exp((x * x)) / fabs(x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (1.0 + ((1.875 * Math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * ((Math.exp((x * x)) / Math.abs(x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (1.0 + ((1.875 * math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * ((math.exp((x * x)) / math.fabs(x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))) * Float64(Float64(exp(Float64(x * x)) / abs(x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((1.875 * (x ^ -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * ((exp((x * x)) / abs(x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{x}^{6}}{1.875}}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. associate-/r/ 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{x}^{6}} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. pow-flip 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{\left(-6\right)}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-6}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-6} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

6. Final simplification 100.0%

   \[\leadsto \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \]

Alternative 5: 99.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (/ (exp (* x x)) (fabs x)) (sqrt PI))
  (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))))))

C

double code(double x) {
	return ((exp((x * x)) / fabs(x)) / sqrt(((double) M_PI))) * (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))));
}

Java

public static double code(double x) {
	return ((Math.exp((x * x)) / Math.abs(x)) / Math.sqrt(Math.PI)) * (1.0 + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))));
}

Python

def code(x):
	return ((math.exp((x * x)) / math.fabs(x)) / math.sqrt(math.pi)) * (1.0 + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))

Julia

function code(x)
	return Float64(Float64(Float64(exp(Float64(x * x)) / abs(x)) / sqrt(pi)) * Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp((x * x)) / abs(x)) / sqrt(pi)) * (1.0 + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))));
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Final simplification 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 6: 52.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (* 1.875 (pow x -6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))
  (/ (/ (fma x x 1.0) x) (sqrt PI))))

C

double code(double x) {
	return (1.0 + ((1.875 * pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * ((fma(x, x, 1.0) / x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))) * Float64(Float64(fma(x, x, 1.0) / x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x + 1.0), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{x}^{6}}{1.875}}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. associate-/r/ 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{x}^{6}} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. pow-flip 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{\left(-6\right)}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-6}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-6} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

6. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

8. Simplified 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 54.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + x \cdot x}{\left|x\right|}\right)\right)}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. expm1-udef 54.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{1 + x \cdot x}{\left|x\right|}\right)} - 1}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. +-commutative 54.7%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{x \cdot x + 1}}{\left|x\right|}\right)} - 1}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. fma-def 54.7%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\left|x\right|}\right)} - 1}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   5. add-sqr-sqrt 54.7%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, 1\right)}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)} - 1}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   6. fabs-sqr 54.7%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)} - 1}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   7. add-sqr-sqrt 54.7%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{x}}\right)} - 1}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

10. Applied egg-rr 54.7%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, 1\right)}{x}\right)} - 1}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
11. Step-by-step derivation

    1. expm1-def 54.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, 1\right)}{x}\right)\right)}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

    2. expm1-log1p 54.7%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

12. Simplified 54.7%

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

13. Final simplification 54.7%

    \[\leadsto \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}{\sqrt{\pi}} \]

Alternative 7: 51.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))))
  (/ (fma x x 1.0) (* x (sqrt PI)))))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) * (fma(x, x, 1.0) / (x * sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) * Float64(fma(x, x, 1.0) / Float64(x * sqrt(pi))))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x + 1.0), $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Simplified 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 54.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + x \cdot x}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-udef 54.7%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot x}{\left|x\right|}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. add-sqr-sqrt 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. fabs-sqr 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. add-sqr-sqrt 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot x}{\color{blue}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. associate-/l/ 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 + x \cdot x}{\sqrt{\pi} \cdot x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. associate-/r* 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1 + x \cdot x}{\sqrt{\pi}}}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. +-commutative 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{x \cdot x + 1}}{\sqrt{\pi}}}{x}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. fma-def 54.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\sqrt{\pi}}}{x}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Applied egg-rr 54.7%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
12. Step-by-step derivation

    1. expm1-def 54.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 54.7%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. associate-/l/ 54.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

13. Simplified 54.7%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Final simplification 54.7%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \]

Alternative 8: 5.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{x}{\frac{x}{x}} + \frac{1}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (* 1.875 (pow x -6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))
  (/ (+ (/ x (/ x x)) (/ 1.0 x)) (sqrt PI))))

C

double code(double x) {
	return (1.0 + ((1.875 * pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * (((x / (x / x)) + (1.0 / x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (1.0 + ((1.875 * Math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * (((x / (x / x)) + (1.0 / x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (1.0 + ((1.875 * math.pow(x, -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * (((x / (x / x)) + (1.0 / x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))) * Float64(Float64(Float64(x / Float64(x / x)) + Float64(1.0 / x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((1.875 * (x ^ -6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) * (((x / (x / x)) + (1.0 / x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x / N[(x / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{x}^{6}}{1.875}}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   2. associate-/r/ 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{x}^{6}} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. pow-flip 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{\left(-6\right)}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-6}} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-6} \cdot 1.875} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

6. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

8. Simplified 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

9. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{\left|x\right|} + \frac{1}{\left|x\right|}}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. unpow2 54.7%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{\left|x\right|} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. associate-/l* 5.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{\frac{\left|x\right|}{x}}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. unpow1 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{1}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. sqr-pow 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. fabs-sqr 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    10. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{0.5}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    11. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    12. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{x}^{0.5}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    13. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{1}{2}\right)}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    14. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{1}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    15. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    16. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{1}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    17. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    18. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    19. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    20. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    21. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Simplified 5.7%

    \[\leadsto \frac{\color{blue}{\frac{x}{\frac{x}{x}} + \frac{1}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left({x}^{-6} \cdot 1.875 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

12. Final simplification 5.7%

    \[\leadsto \left(1 + \left(1.875 \cdot {x}^{-6} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \cdot \frac{\frac{x}{\frac{x}{x}} + \frac{1}{x}}{\sqrt{\pi}} \]

Alternative 9: 5.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{x + \frac{1}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))))
  (/ (+ x (/ 1.0 x)) (sqrt PI))))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) * ((x + (1.0 / x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))) * ((x + (1.0 / x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (1.0 + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))) * ((x + (1.0 / x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) * Float64(Float64(x + Float64(1.0 / x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))) * ((x + (1.0 / x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Simplified 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

10. Taylor expanded in x around 0 54.7%

    \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{\left|x\right|} + \frac{1}{\left|x\right|}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 54.7%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{\left|x\right|} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. associate-/l* 5.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{\frac{\left|x\right|}{x}}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. unpow1 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{1}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. sqr-pow 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. fabs-sqr 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    10. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{0.5}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    11. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    12. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{x}^{0.5}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    13. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{1}{2}\right)}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    14. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{1}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    15. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    16. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{1}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    17. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    18. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    19. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    20. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    21. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Simplified 5.7%

    \[\leadsto \frac{\color{blue}{\frac{x}{\frac{x}{x}} + \frac{1}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
13. Step-by-step derivation

    1. expm1-log1p-u 5.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{\frac{x}{x}} + \frac{1}{x}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-udef 5.7%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{x}{\frac{x}{x}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. associate-/r/ 5.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{x} \cdot x} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. *-inverses 5.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1} \cdot x + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. *-un-lft-identity 5.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Applied egg-rr 5.7%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
15. Step-by-step derivation

    1. expm1-def 5.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + \frac{1}{x}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 5.7%

       \[\leadsto \color{blue}{\frac{x + \frac{1}{x}}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

16. Simplified 5.7%

    \[\leadsto \color{blue}{\frac{x + \frac{1}{x}}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

17. Final simplification 5.7%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{x + \frac{1}{x}}{\sqrt{\pi}} \]

Alternative 10: 5.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))) (* x (sqrt (/ 1.0 PI)))))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) * (x * sqrt((1.0 / ((double) M_PI))));
}

Java

public static double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))) * (x * Math.sqrt((1.0 / Math.PI)));
}

Python

def code(x):
	return (1.0 + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))) * (x * math.sqrt((1.0 / math.pi)))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) * Float64(x * sqrt(Float64(1.0 / pi))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))) * (x * sqrt((1.0 / pi)));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Simplified 54.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

10. Taylor expanded in x around 0 54.7%

    \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{\left|x\right|} + \frac{1}{\left|x\right|}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 54.7%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{\left|x\right|} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. associate-/l* 5.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{\frac{\left|x\right|}{x}}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. unpow1 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{1}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. sqr-pow 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. metadata-eval 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. unpow1/2 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. fabs-sqr 5.7%

       \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    10. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{0.5}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    11. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    12. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{x}^{0.5}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    13. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{1}{2}\right)}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    14. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{{x}^{1}}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    15. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{\color{blue}{x}}{x}} + \frac{1}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    16. unpow1 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{1}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    17. sqr-pow 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    18. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    19. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    20. metadata-eval 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    21. unpow1/2 5.7%

        \[\leadsto \frac{\frac{x}{\frac{x}{x}} + \frac{1}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Simplified 5.7%

    \[\leadsto \frac{\color{blue}{\frac{x}{\frac{x}{x}} + \frac{1}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

13. Taylor expanded in x around inf 5.7%

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot x\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Final simplification 5.7%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

Alternative 11: 2.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))) (/ (pow PI -0.5) x)))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) * (pow(((double) M_PI), -0.5) / x);
}

Java

public static double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))) * (Math.pow(Math.PI, -0.5) / x);
}

Python

def code(x):
	return (1.0 + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))) * (math.pow(math.pi, -0.5) / x)

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) * Float64((pi ^ -0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))) * ((pi ^ -0.5) / x);
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 2.4%

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 2.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   2. expm1-udef 1.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   3. div-inv 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\left|x\right|} \cdot \frac{1}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   4. clear-num 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\left|x\right|}{1}}} \cdot \frac{1}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   5. associate-*l/ 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{1}{\sqrt{\pi}}}{\frac{\left|x\right|}{1}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   6. *-un-lft-identity 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{\sqrt{\pi}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   7. pow1/2 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   8. pow-flip 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   9. metadata-eval 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   10. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   11. fabs-sqr 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   12. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   13. /-rgt-identity 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\color{blue}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Applied egg-rr 1.8%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Final simplification 2.4%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{{\pi}^{-0.5}}{x} \]

Alternative 12: 2.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (/ 1.0 x) (/ 0.5 (pow x 3.0)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((1.0 / x) + (0.5 / pow(x, 3.0)));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * ((1.0 / x) + (0.5 / Math.pow(x, 3.0)));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((1.0 / x) + (0.5 / math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(1.0 / x) + Float64(0.5 / (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((1.0 / x) + (0.5 / (x ^ 3.0)));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 2.4%

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 2.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   2. expm1-udef 1.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   3. div-inv 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\left|x\right|} \cdot \frac{1}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   4. clear-num 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\left|x\right|}{1}}} \cdot \frac{1}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   5. associate-*l/ 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{1}{\sqrt{\pi}}}{\frac{\left|x\right|}{1}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   6. *-un-lft-identity 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{\sqrt{\pi}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   7. pow1/2 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   8. pow-flip 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   9. metadata-eval 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   10. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   11. fabs-sqr 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   12. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   13. /-rgt-identity 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\color{blue}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Applied egg-rr 1.8%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Taylor expanded in x around inf 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{1}{x} + 0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. *-commutative 2.4%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}} + 0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) \]

    2. associate-*r* 2.4%

       \[\leadsto \frac{1}{x} \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    3. distribute-rgt-out 2.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

    4. associate-*r/ 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

    5. metadata-eval 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

14. Simplified 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

15. Final simplification 2.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) \]

Alternative 13: 1.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{1.875}{{x}^{7}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (/ 1.875 (pow x 7.0))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (1.875 / pow(x, 7.0));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (1.875 / Math.pow(x, 7.0));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (1.875 / math.pow(x, 7.0))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(1.875 / (x ^ 7.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (1.875 / (x ^ 7.0));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 2.4%

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 2.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   2. expm1-udef 1.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   3. div-inv 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\left|x\right|} \cdot \frac{1}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   4. clear-num 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\left|x\right|}{1}}} \cdot \frac{1}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   5. associate-*l/ 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{1}{\sqrt{\pi}}}{\frac{\left|x\right|}{1}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   6. *-un-lft-identity 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{\sqrt{\pi}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   7. pow1/2 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   8. pow-flip 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   9. metadata-eval 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   10. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   11. fabs-sqr 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   12. add-sqr-sqrt 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   13. /-rgt-identity 1.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\color{blue}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Applied egg-rr 1.8%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)} - 1\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Taylor expanded in x around 0 1.8%

    \[\leadsto \color{blue}{1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. associate-*r* 1.8%

       \[\leadsto \color{blue}{\left(1.875 \cdot \frac{1}{{x}^{7}}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    2. associate-*r/ 1.8%

       \[\leadsto \color{blue}{\frac{1.875 \cdot 1}{{x}^{7}}} \cdot \sqrt{\frac{1}{\pi}} \]

    3. metadata-eval 1.8%

       \[\leadsto \frac{\color{blue}{1.875}}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}} \]

14. Simplified 1.8%

    \[\leadsto \color{blue}{\frac{1.875}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}} \]

15. Final simplification 1.8%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{1.875}{{x}^{7}} \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))