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

Percentage Accurate: 100.0% → 100.0%

Time: 15.3s

Alternatives: 6

Speedup: 2.4×

• 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, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x)))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]
4. Step-by-step derivation

   1. add-log-exp 99.9%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. *-un-lft-identity 99.9%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. log-prod 99.9%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 99.9%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. log-prod 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

Alternative 2: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\frac{\sqrt{\pi}}{\frac{1}{x}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (/ (sqrt PI) (/ 1.0 x)))
  (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / (sqrt(((double) M_PI)) / (1.0 / x))) * (1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (Math.sqrt(Math.PI) / (1.0 / x))) * (1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / (math.sqrt(math.pi) / (1.0 / x))) * (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(sqrt(pi) / Float64(1.0 / x))) * Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / (sqrt(pi) / (1.0 / x))) * (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(1 \cdot e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log 1 + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{0} + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{x}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(0 + \frac{0.75}{{x}^{4}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

   1. add-cube-cbrt 99.9%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\left|x\right| \cdot \sqrt{\pi}} \cdot \sqrt[3]{\left|x\right| \cdot \sqrt{\pi}}\right) \cdot \sqrt[3]{\left|x\right| \cdot \sqrt{\pi}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. pow3 99.9%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot \sqrt{\pi}}\right)}^{3}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}}\right)}^{3}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}}\right)}^{3}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-sqr-sqrt 99.9%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\color{blue}{x} \cdot \sqrt{\pi}}\right)}^{3}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. *-commutative 99.9%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot x}}\right)}^{3}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi} \cdot x}\right)}^{3}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. rem-cube-cbrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    4. fabs-sqr 100.0%

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

    5. add-sqr-sqrt 100.0%

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

    6. /-rgt-identity 100.0%

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

    7. add-sqr-sqrt 100.0%

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

    8. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\frac{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    9. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\frac{\color{blue}{x} \cdot \sqrt{\pi}}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    10. *-commutative 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\frac{\color{blue}{\sqrt{\pi} \cdot x}}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    11. associate-/l* 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\frac{\sqrt{\pi}}{\frac{1}{x}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\frac{\sqrt{\pi}}{\frac{1}{x}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\frac{\sqrt{\pi}}{\frac{1}{x}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

Alternative 3: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
  (/ (exp (pow x 2.0)) (* x (sqrt PI)))))

C

double code(double x) {
	return (1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) * (exp(pow(x, 2.0)) / (x * sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	return (1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) * (Math.exp(Math.pow(x, 2.0)) / (x * Math.sqrt(Math.PI)));
}

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (math.exp(math.pow(x, 2.0)) / (x * math.sqrt(math.pi)))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) * Float64(exp((x ^ 2.0)) / Float64(x * sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * (exp((x ^ 2.0)) / (x * sqrt(pi)));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(1 \cdot e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log 1 + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{0} + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{x}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(0 + \frac{0.75}{{x}^{4}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}}}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

9. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{\sqrt{\pi} \cdot \left|x\right|}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. unpow1 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{1}}\right|} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. sqr-pow 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. fabs-sqr 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    4. sqr-pow 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{{x}^{1}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    5. unpow1 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{x}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    6. *-commutative 99.9%

       \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

11. Simplified 99.9%

    \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

12. Final simplification 99.9%

    \[\leadsto \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}} \]

Alternative 4: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
  (/ (pow (exp x) x) (* x (sqrt PI)))))

C

double code(double x) {
	return (1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) * (pow(exp(x), x) / (x * sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	return (1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) * (Math.pow(Math.exp(x), x) / (x * Math.sqrt(Math.PI)));
}

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (math.pow(math.exp(x), x) / (x * math.sqrt(math.pi)))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) * Float64((exp(x) ^ x) / Float64(x * sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * ((exp(x) ^ x) / (x * sqrt(pi)));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(1 \cdot e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log 1 + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{0} + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{x}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(0 + \frac{0.75}{{x}^{4}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 100.0%

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

   1. unpow1 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{1}}\right|} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. sqr-pow 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. fabs-sqr 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{{x}^{1}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. unpow1 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \color{blue}{x}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. *-commutative 99.9%

      \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \]

Alternative 5: 5.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{1}{\frac{1}{x \cdot \sqrt{\pi}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
  (/ 1.0 (/ 1.0 (* x (sqrt PI))))))

C

double code(double x) {
	return (1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) * (1.0 / (1.0 / (x * sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return (1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) * (1.0 / (1.0 / (x * Math.sqrt(Math.PI))));
}

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (1.0 / (1.0 / (x * math.sqrt(math.pi))))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) * Float64(1.0 / Float64(1.0 / Float64(x * sqrt(pi)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * (1.0 / (1.0 / (x * sqrt(pi))));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(1 \cdot e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log 1 + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{0} + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{x}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(0 + \frac{0.75}{{x}^{4}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 2.4%

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

   1. expm1-log1p-u 2.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot \sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. expm1-udef 1.8%

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

   3. associate-/r* 1.8%

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

   4. add-sqr-sqrt 1.8%

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

   5. fabs-sqr 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

10. Applied egg-rr 1.8%

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

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{x}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. associate-/r* 2.4%

       \[\leadsto \color{blue}{\frac{1}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

12. Simplified 2.4%

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

    1. add-exp-log 2.4%

       \[\leadsto \frac{1}{\color{blue}{e^{\log \left(x \cdot \sqrt{\pi}\right)}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. add-sqr-sqrt 2.4%

       \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\log \left(x \cdot \sqrt{\pi}\right)} \cdot \sqrt{\log \left(x \cdot \sqrt{\pi}\right)}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. sqrt-unprod 2.4%

       \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\log \left(x \cdot \sqrt{\pi}\right) \cdot \log \left(x \cdot \sqrt{\pi}\right)}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    4. sqr-neg 2.4%

       \[\leadsto \frac{1}{e^{\sqrt{\color{blue}{\left(-\log \left(x \cdot \sqrt{\pi}\right)\right) \cdot \left(-\log \left(x \cdot \sqrt{\pi}\right)\right)}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    5. sqrt-unprod 0.0%

       \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{-\log \left(x \cdot \sqrt{\pi}\right)} \cdot \sqrt{-\log \left(x \cdot \sqrt{\pi}\right)}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    6. add-sqr-sqrt 5.7%

       \[\leadsto \frac{1}{e^{\color{blue}{-\log \left(x \cdot \sqrt{\pi}\right)}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    7. exp-neg 5.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{\log \left(x \cdot \sqrt{\pi}\right)}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    8. add-exp-log 5.7%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sqrt{\pi}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    9. *-commutative 5.7%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\sqrt{\pi} \cdot x}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

14. Applied egg-rr 5.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sqrt{\pi} \cdot x}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

15. Final simplification 5.7%

    \[\leadsto \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{1}{\frac{1}{x \cdot \sqrt{\pi}}} \]

Alternative 6: 3.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))))))

C

double code(double x) {
	return 1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((0.75d0 / (x ** 4.0d0)) + ((0.5d0 / (x * x)) + (1.875d0 / (x ** 6.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))));
}

Python

def code(x):
	return 1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + 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 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(1 \cdot e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log 1 + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{0} + \log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(0 + \frac{0.75}{{\color{blue}{x}}^{4}}\right) + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(0 + \frac{0.75}{{x}^{4}}\right)} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 2.4%

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

   1. expm1-log1p-u 2.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot \sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. expm1-udef 1.8%

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

   3. associate-/r* 1.8%

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

   4. add-sqr-sqrt 1.8%

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

   5. fabs-sqr 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 1.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

10. Applied egg-rr 1.8%

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

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{x}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. associate-/r* 2.4%

       \[\leadsto \color{blue}{\frac{1}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

12. Simplified 2.4%

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

    1. /-rgt-identity 2.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \sqrt{\pi}}{1}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    2. add-sqr-sqrt 2.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{x \cdot \sqrt{\pi}} \cdot \sqrt{x \cdot \sqrt{\pi}}}}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    3. associate-/l* 2.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{x \cdot \sqrt{\pi}}}{\frac{1}{\sqrt{x \cdot \sqrt{\pi}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    4. *-commutative 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sqrt{\pi} \cdot x}}}{\frac{1}{\sqrt{x \cdot \sqrt{\pi}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    5. metadata-eval 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x \cdot \sqrt{\pi}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    6. sqrt-div 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\color{blue}{\sqrt{\frac{1}{x \cdot \sqrt{\pi}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    7. inv-pow 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{\color{blue}{{\left(x \cdot \sqrt{\pi}\right)}^{-1}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    8. pow-to-exp 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{\color{blue}{e^{\log \left(x \cdot \sqrt{\pi}\right) \cdot -1}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    9. *-commutative 2.4%

       \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{-1 \cdot \log \left(x \cdot \sqrt{\pi}\right)}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    10. neg-mul-1 2.4%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{-\log \left(x \cdot \sqrt{\pi}\right)}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    11. add-sqr-sqrt 0.0%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{\sqrt{-\log \left(x \cdot \sqrt{\pi}\right)} \cdot \sqrt{-\log \left(x \cdot \sqrt{\pi}\right)}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    12. sqrt-unprod 3.3%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{\sqrt{\left(-\log \left(x \cdot \sqrt{\pi}\right)\right) \cdot \left(-\log \left(x \cdot \sqrt{\pi}\right)\right)}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    13. sqr-neg 3.3%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\sqrt{\color{blue}{\log \left(x \cdot \sqrt{\pi}\right) \cdot \log \left(x \cdot \sqrt{\pi}\right)}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    14. sqrt-unprod 3.3%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{\sqrt{\log \left(x \cdot \sqrt{\pi}\right)} \cdot \sqrt{\log \left(x \cdot \sqrt{\pi}\right)}}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    15. add-sqr-sqrt 3.3%

        \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{\pi} \cdot x}}{\sqrt{e^{\color{blue}{\log \left(x \cdot \sqrt{\pi}\right)}}}}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

14. Applied egg-rr 3.3%

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

    1. *-inverses 3.3%

       \[\leadsto \frac{1}{\color{blue}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

16. Simplified 3.3%

    \[\leadsto \frac{1}{\color{blue}{1}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

17. Final simplification 3.3%

    \[\leadsto 1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \]

Reproduce

herbie shell --seed 2023312 
(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)))))))