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

Percentage Accurate: 100.0% → 100.0%

Time: 18.1s

Alternatives: 8

Speedup: 4.2×

• 99.4% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \log \left(1 + \mathsf{expm1}\left(\frac{\frac{0.5}{x}}{x}\right)\right), \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (pow (pow PI 0.25) 2.0))
  (fma
   (+ 1.0 (log (+ 1.0 (expm1 (/ (/ 0.5 x) x)))))
   (/ 1.0 (fabs x))
   (log
    (+ 1.0 (expm1 (/ (fma 1.875 (pow x -2.0) 0.75) (/ x (pow x -4.0)))))))))

C

double code(double x) {
	return (pow(exp(x), x) / pow(pow(((double) M_PI), 0.25), 2.0)) * fma((1.0 + log((1.0 + expm1(((0.5 / x) / x))))), (1.0 / fabs(x)), log((1.0 + expm1((fma(1.875, pow(x, -2.0), 0.75) / (x / pow(x, -4.0)))))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / ((pi ^ 0.25) ^ 2.0)) * fma(Float64(1.0 + log(Float64(1.0 + expm1(Float64(Float64(0.5 / x) / x))))), Float64(1.0 / abs(x)), log(Float64(1.0 + expm1(Float64(fma(1.875, (x ^ -2.0), 0.75) / Float64(x / (x ^ -4.0))))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Log[N[(1.0 + N[(Exp[N[(N[(1.875 * N[Power[x, -2.0], $MachinePrecision] + 0.75), $MachinePrecision] / N[(x / N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
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. log1p-expm1-u 100.0%

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

   2. log1p-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. clear-num 100.0%

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

   5. un-div-inv 100.0%

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

   6. +-commutative 100.0%

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

   7. div-inv 100.0%

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

   8. fma-def 100.0%

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

   9. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \frac{1}{\color{blue}{{x}^{2}}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \color{blue}{{x}^{\left(-2\right)}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{\color{blue}{-2}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{{x}^{-4}}}\right)\right)\right) \]

   13. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{{x}^{-4}}}\right)\right)\right) \]

   14. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{x}}{{x}^{-4}}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)}\right) \]
10. Step-by-step derivation

    1. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    2. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt{\sqrt{\pi}}\right)}^{2}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    3. pow1/2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt{\color{blue}{{\pi}^{0.5}}}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    4. sqrt-pow1 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.25}}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]
12. Step-by-step derivation

    1. log1p-expm1-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    2. log1p-udef 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    3. associate-/r* 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{\frac{0.5}{x}}{x}}\right)\right), \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

13. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\frac{0.5}{x}}{x}\right)\right)}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

14. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \log \left(1 + \mathsf{expm1}\left(\frac{\frac{0.5}{x}}{x}\right)\right), \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (pow (pow PI 0.25) 2.0))
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (log
    (+ 1.0 (expm1 (/ (fma 1.875 (pow x -2.0) 0.75) (/ x (pow x -4.0)))))))))

C

double code(double x) {
	return (pow(exp(x), x) / pow(pow(((double) M_PI), 0.25), 2.0)) * fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), log((1.0 + expm1((fma(1.875, pow(x, -2.0), 0.75) / (x / pow(x, -4.0)))))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / ((pi ^ 0.25) ^ 2.0)) * fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), log(Float64(1.0 + expm1(Float64(fma(1.875, (x ^ -2.0), 0.75) / Float64(x / (x ^ -4.0))))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Log[N[(1.0 + N[(Exp[N[(N[(1.875 * N[Power[x, -2.0], $MachinePrecision] + 0.75), $MachinePrecision] / N[(x / N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
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. log1p-expm1-u 100.0%

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

   2. log1p-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. clear-num 100.0%

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

   5. un-div-inv 100.0%

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

   6. +-commutative 100.0%

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

   7. div-inv 100.0%

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

   8. fma-def 100.0%

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

   9. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \frac{1}{\color{blue}{{x}^{2}}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \color{blue}{{x}^{\left(-2\right)}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{\color{blue}{-2}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{{x}^{-4}}}\right)\right)\right) \]

   13. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{{x}^{-4}}}\right)\right)\right) \]

   14. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{x}}{{x}^{-4}}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)}\right) \]
10. Step-by-step derivation

    1. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    2. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt{\sqrt{\pi}}\right)}^{2}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    3. pow1/2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt{\color{blue}{{\pi}^{0.5}}}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    4. sqrt-pow1 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.25}}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right) \]

Alternative 3: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (log (+ 1.0 (expm1 (* (fma 1.875 (pow x -2.0) 0.75) (pow x -5.0))))))))

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)), log((1.0 + expm1((fma(1.875, pow(x, -2.0), 0.75) * pow(x, -5.0))))));
}

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)), log(Float64(1.0 + expm1(Float64(fma(1.875, (x ^ -2.0), 0.75) * (x ^ -5.0)))))))
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[Log[N[(1.0 + N[(Exp[N[(N[(1.875 * N[Power[x, -2.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]] - 1), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
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. log1p-expm1-u 100.0%

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

   2. log1p-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. clear-num 100.0%

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

   5. un-div-inv 100.0%

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

   6. +-commutative 100.0%

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

   7. div-inv 100.0%

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

   8. fma-def 100.0%

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

   9. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \frac{1}{\color{blue}{{x}^{2}}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \color{blue}{{x}^{\left(-2\right)}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{\color{blue}{-2}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{{x}^{-4}}}\right)\right)\right) \]

   13. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{{x}^{-4}}}\right)\right)\right) \]

   14. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{x}}{{x}^{-4}}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)}\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right)\right)}\right) \]

    2. expm1-udef 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)\right)} - 1}\right) \]

    3. log1p-def 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)}\right)} - 1\right) \]

    4. log1p-expm1-u 100.0%

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

    5. div-inv 100.0%

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

    6. clear-num 100.0%

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

    7. pow1 100.0%

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

    8. pow-div 100.0%

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

    9. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{\color{blue}{-5}}\right)} - 1\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)} - 1}\right) \]
12. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)}\right) \]

    2. expm1-log1p-u 100.0%

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

    3. log1p-expm1-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)}\right) \]

    4. log1p-udef 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)}\right) \]

13. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)}\right) \]

14. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right) \cdot {x}^{-5}\right)\right)\right) \]

Alternative 4: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

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)), ((0.75 / pow(x, 5.0)) + (1.875 / pow(x, 7.0))));
}

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(0.75 / (x ^ 5.0)) + Float64(1.875 / (x ^ 7.0)))))
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[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
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. log1p-expm1-u 100.0%

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

   2. log1p-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. clear-num 100.0%

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

   5. un-div-inv 100.0%

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

   6. +-commutative 100.0%

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

   7. div-inv 100.0%

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

   8. fma-def 100.0%

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

   9. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \frac{1}{\color{blue}{{x}^{2}}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, \color{blue}{{x}^{\left(-2\right)}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{\color{blue}{-2}}, 0.75\right)}{\frac{\left|x\right|}{{x}^{-4}}}\right)\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{{x}^{-4}}}\right)\right)\right) \]

   13. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{{x}^{-4}}}\right)\right)\right) \]

   14. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{\color{blue}{x}}{{x}^{-4}}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)}{\frac{x}{{x}^{-4}}}\right)\right)}\right) \]

10. Taylor expanded in x around inf 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. associate-*r/ 100.0%

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

    4. metadata-eval 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 5: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (expm1 (log1p (* x (sqrt PI)))))
  (+
   1.0
   (+ (/ 1.875 (pow (fabs x) 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))))

C

double code(double x) {
	return (exp((x * x)) / expm1(log1p((x * sqrt(((double) M_PI)))))) * (1.0 + ((1.875 / pow(fabs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.expm1(Math.log1p((x * Math.sqrt(Math.PI))))) * (1.0 + ((1.875 / Math.pow(Math.abs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Python

def code(x):
	return (math.exp((x * x)) / math.expm1(math.log1p((x * math.sqrt(math.pi))))) * (1.0 + ((1.875 / math.pow(math.fabs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / expm1(log1p(Float64(x * sqrt(pi))))) * Float64(1.0 + Float64(Float64(1.875 / (abs(x) ^ 6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(Exp[N[Log[1 + N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 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 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 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. *-commutative 99.9%

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

   3. add-sqr-sqrt 99.9%

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

   4. fabs-sqr 99.9%

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

   5. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{x \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (* x (pow (cbrt (sqrt PI)) 3.0)))
  (+ (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)) (+ 1.0 (/ 1.875 (pow x 6.0))))))

C

double code(double x) {
	return (exp((x * x)) / (x * pow(cbrt(sqrt(((double) M_PI))), 3.0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / pow(x, 6.0))));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / (x * Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / Math.pow(x, 6.0))));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / Float64(x * (cbrt(sqrt(pi)) ^ 3.0))) * Float64(Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)) + Float64(1.0 + Float64(1.875 / (x ^ 6.0)))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + 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 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

   3. *-commutative 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

7. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 7: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.pow(x, -5.0) * (0.75 + (1.875 / (x * x)))));
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (math.pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * (((1.0 + (0.5 / (x * x))) / abs(x)) + ((x ^ -5.0) * (0.75 + (1.875 / (x * x)))));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $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 99.9%

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

   1. expm1-log1p-u 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)\right)} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-udef 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)} - 1\right)} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   3. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   4. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   5. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   6. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   7. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   8. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-5}}\right)} - 1\right) \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5}\right)} - 1\right)} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-5}\right)\right)} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-log1p 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 8: 99.9% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $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 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

   3. *-commutative 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Reproduce

herbie shell --seed 2023238 
(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)))))))