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

Percentage Accurate: 100.0% → 100.0%

Time: 22.2s

Alternatives: 7

Speedup: 3.3×

• 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, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) + -1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ 1.875 (pow x 7.0))
   (fma 0.75 (+ (+ 1.0 (pow x -5.0)) -1.0) (fma 0.5 (pow x -3.0) (/ 1.0 x))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((1.875 / pow(x, 7.0)) + fma(0.75, ((1.0 + pow(x, -5.0)) + -1.0), fma(0.5, pow(x, -3.0), (1.0 / x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(1.875 / (x ^ 7.0)) + fma(0.75, Float64(Float64(1.0 + (x ^ -5.0)) + -1.0), fma(0.5, (x ^ -3.0), Float64(1.0 / x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[(N[(1.0 + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(1.0 / 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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. associate-*r/ 100.0%

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

   3. pow-prod-up 100.0%

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

   4. inv-pow 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow-pow 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. metadata-eval 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)\right)}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   2. expm1-define 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)} - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. pow1 100.0%

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

   6. pow-div 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \left(1 + {x}^{\color{blue}{-5}}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\left(1 + {x}^{-5}\right) - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{1.875}{{x}^{7}}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

10. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) + -1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{1 + \left(\frac{0.5}{{x}^{2}} + \frac{0.75}{{x}^{4}}\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ 1.875 (pow x 7.0))
   (/ (+ 1.0 (+ (/ 0.5 (pow x 2.0)) (/ 0.75 (pow x 4.0)))) x))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((1.875 / pow(x, 7.0)) + ((1.0 + ((0.5 / pow(x, 2.0)) + (0.75 / pow(x, 4.0)))) / x));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((1.875 / Math.pow(x, 7.0)) + ((1.0 + ((0.5 / Math.pow(x, 2.0)) + (0.75 / Math.pow(x, 4.0)))) / x));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((1.875 / math.pow(x, 7.0)) + ((1.0 + ((0.5 / math.pow(x, 2.0)) + (0.75 / math.pow(x, 4.0)))) / x))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(1.875 / (x ^ 7.0)) + Float64(Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(0.75 / (x ^ 4.0)))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((1.875 / (x ^ 7.0)) + ((1.0 + ((0.5 / (x ^ 2.0)) + (0.75 / (x ^ 4.0)))) / x));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. associate-*r/ 100.0%

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

   3. pow-prod-up 100.0%

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

   4. inv-pow 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow-pow 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. metadata-eval 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)\right)}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   2. expm1-define 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)} - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. pow1 100.0%

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

   6. pow-div 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \left(1 + {x}^{\color{blue}{-5}}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\left(1 + {x}^{-5}\right) - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{1.875}{{x}^{7}}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

10. Taylor expanded in x around inf 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{x}}\right) \]
11. Step-by-step derivation

    1. +-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \frac{0.75}{{x}^{4}}\right)}}{x}\right) \]

    2. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \frac{0.75}{{x}^{4}}\right)}{x}\right) \]

    3. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \frac{0.75}{{x}^{4}}\right)}{x}\right) \]

12. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\frac{1 + \left(\frac{0.5}{{x}^{2}} + \frac{0.75}{{x}^{4}}\right)}{x}}\right) \]
13. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \frac{0.75 + \frac{1.875}{{x}^{2}}}{{x}^{4}}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (* 0.5 (pow x -3.0))
   (/ (+ 1.0 (/ (+ 0.75 (/ 1.875 (pow x 2.0))) (pow x 4.0))) x))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((0.5 * pow(x, -3.0)) + ((1.0 + ((0.75 + (1.875 / pow(x, 2.0))) / pow(x, 4.0))) / x));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((0.5 * Math.pow(x, -3.0)) + ((1.0 + ((0.75 + (1.875 / Math.pow(x, 2.0))) / Math.pow(x, 4.0))) / x));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((0.5 * math.pow(x, -3.0)) + ((1.0 + ((0.75 + (1.875 / math.pow(x, 2.0))) / math.pow(x, 4.0))) / x))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(0.5 * (x ^ -3.0)) + Float64(Float64(1.0 + Float64(Float64(0.75 + Float64(1.875 / (x ^ 2.0))) / (x ^ 4.0))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((0.5 * (x ^ -3.0)) + ((1.0 + ((0.75 + (1.875 / (x ^ 2.0))) / (x ^ 4.0))) / x));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.75 + N[(1.875 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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%

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

   1. fma-undefine 100.0%

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

   2. pow-flip 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l/ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \color{blue}{\frac{0.75 + 1.875 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}}{x}\right) \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \frac{0.75 + \color{blue}{\frac{1.875 \cdot 1}{{x}^{2}}}}{{x}^{4}}}{x}\right) \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \frac{0.75 + \frac{\color{blue}{1.875}}{{x}^{2}}}{{x}^{4}}}{x}\right) \]

9. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \color{blue}{\frac{0.75 + \frac{1.875}{{x}^{2}}}{{x}^{4}}}}{x}\right) \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+ (* 0.5 (pow x -3.0)) (/ (+ 1.0 (/ 0.75 (pow x 4.0))) x))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((0.5 * pow(x, -3.0)) + ((1.0 + (0.75 / pow(x, 4.0))) / x));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((0.5 * Math.pow(x, -3.0)) + ((1.0 + (0.75 / Math.pow(x, 4.0))) / x));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((0.5 * math.pow(x, -3.0)) + ((1.0 + (0.75 / math.pow(x, 4.0))) / x))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(0.5 * (x ^ -3.0)) + Float64(Float64(1.0 + Float64(0.75 / (x ^ 4.0))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((0.5 * (x ^ -3.0)) + ((1.0 + (0.75 / (x ^ 4.0))) / x));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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%

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

   1. fma-undefine 100.0%

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

   2. pow-flip 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l/ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \color{blue}{\frac{0.75}{{x}^{4}}}}{x}\right) \]
8. Add Preprocessing

Alternative 5: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((1.0 + (0.5 / (x ^ 2.0))) / x);
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[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. associate-*r/ 100.0%

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

   3. pow-prod-up 100.0%

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

   4. inv-pow 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow-pow 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. metadata-eval 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)\right)}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   2. expm1-define 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)} - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. pow1 100.0%

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

   6. pow-div 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \left(1 + {x}^{\color{blue}{-5}}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\left(1 + {x}^{-5}\right) - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{1.875}{{x}^{7}}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

10. Taylor expanded in x around inf 99.3%

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

    1. associate-*r/ 99.3%

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

    2. metadata-eval 99.3%

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

12. Simplified 99.3%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}} \]
13. Add Preprocessing

Alternative 6: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. associate-*r/ 100.0%

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

   3. pow-prod-up 100.0%

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

   4. inv-pow 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow-pow 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. metadata-eval 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)\right)}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   2. expm1-define 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)} - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. pow1 100.0%

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

   6. pow-div 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \left(1 + {x}^{\color{blue}{-5}}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(1.875 \cdot \frac{{x}^{-6}}{x} + \mathsf{fma}\left(0.75, \color{blue}{\left(1 + {x}^{-5}\right) - 1}, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{1.875}{{x}^{7}}} + \mathsf{fma}\left(0.75, \left(1 + {x}^{-5}\right) - 1, \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)\right) \]

10. Taylor expanded in x around inf 99.2%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 7: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.75 \cdot \left({x}^{-3} \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.75 (* (pow x -3.0) (pow PI -0.5))))

C

double code(double x) {
	return 0.75 * (pow(x, -3.0) * pow(((double) M_PI), -0.5));
}

Java

public static double code(double x) {
	return 0.75 * (Math.pow(x, -3.0) * Math.pow(Math.PI, -0.5));
}

Python

def code(x):
	return 0.75 * (math.pow(x, -3.0) * math.pow(math.pi, -0.5))

Julia

function code(x)
	return Float64(0.75 * Float64((x ^ -3.0) * (pi ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 0.75 * ((x ^ -3.0) * (pi ^ -0.5));
end

Wolfram

code[x_] := N[(0.75 * N[(N[Power[x, -3.0], $MachinePrecision] * N[Power[Pi, -0.5], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}\right)\right)}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   2. expm1-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}\right)} - 1}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{1}{\left|x\right|}}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   4. pow-plus 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)}}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   5. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 1\right)}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{\left({\left(\left|x\right|\right)}^{-1}\right)}^{\color{blue}{4}}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 4\right)}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 4\right)}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{\color{blue}{x}}^{\left(-1 \cdot 4\right)}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{x}^{\color{blue}{-4}}}{\left|x\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   13. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

   14. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{\color{blue}{x}}\right)} - 1, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x}\right)} - 1}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]

7. Taylor expanded in x around 0 1.1%

   \[\leadsto \color{blue}{\frac{0.75 \cdot \sqrt{\frac{1}{\pi}} + 0.75 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)}{{x}^{5}}} \]
8. Step-by-step derivation

   1. distribute-lft-out 1.1%

      \[\leadsto \frac{\color{blue}{0.75 \cdot \left(\sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)}}{{x}^{5}} \]

   2. associate-/l* 1.1%

      \[\leadsto \color{blue}{0.75 \cdot \frac{\sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \sqrt{\frac{1}{\pi}}}{{x}^{5}}} \]

   3. distribute-rgt1-in 1.1%

      \[\leadsto 0.75 \cdot \frac{\color{blue}{\left({x}^{2} + 1\right) \cdot \sqrt{\frac{1}{\pi}}}}{{x}^{5}} \]

9. Simplified 1.1%

   \[\leadsto \color{blue}{0.75 \cdot \frac{\left({x}^{2} + 1\right) \cdot \sqrt{\frac{1}{\pi}}}{{x}^{5}}} \]

10. Taylor expanded in x around inf 1.9%

    \[\leadsto 0.75 \cdot \color{blue}{\left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. *-commutative 1.9%

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

    2. unpow-1 1.9%

       \[\leadsto 0.75 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \frac{1}{{x}^{3}}\right) \]

    3. exp-to-pow 1.9%

       \[\leadsto 0.75 \cdot \left(\sqrt{\color{blue}{e^{\log \pi \cdot -1}}} \cdot \frac{1}{{x}^{3}}\right) \]

    4. *-commutative 1.9%

       \[\leadsto 0.75 \cdot \left(\sqrt{e^{\color{blue}{-1 \cdot \log \pi}}} \cdot \frac{1}{{x}^{3}}\right) \]

    5. neg-mul-1 1.9%

       \[\leadsto 0.75 \cdot \left(\sqrt{e^{\color{blue}{-\log \pi}}} \cdot \frac{1}{{x}^{3}}\right) \]

    6. unpow1/2 1.9%

       \[\leadsto 0.75 \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot \frac{1}{{x}^{3}}\right) \]

    7. exp-prod 1.9%

       \[\leadsto 0.75 \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot \frac{1}{{x}^{3}}\right) \]

    8. distribute-lft-neg-out 1.9%

       \[\leadsto 0.75 \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot \frac{1}{{x}^{3}}\right) \]

    9. distribute-rgt-neg-in 1.9%

       \[\leadsto 0.75 \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot \frac{1}{{x}^{3}}\right) \]

    10. metadata-eval 1.9%

        \[\leadsto 0.75 \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot \frac{1}{{x}^{3}}\right) \]

    11. exp-to-pow 1.9%

        \[\leadsto 0.75 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \frac{1}{{x}^{3}}\right) \]

    12. exp-to-pow 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot \frac{1}{\color{blue}{e^{\log x \cdot 3}}}\right) \]

    13. *-commutative 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot \frac{1}{e^{\color{blue}{3 \cdot \log x}}}\right) \]

    14. exp-neg 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{e^{-3 \cdot \log x}}\right) \]

    15. distribute-lft-neg-in 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot e^{\color{blue}{\left(-3\right) \cdot \log x}}\right) \]

    16. metadata-eval 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot e^{\color{blue}{-3} \cdot \log x}\right) \]

    17. *-commutative 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot e^{\color{blue}{\log x \cdot -3}}\right) \]

    18. exp-to-pow 1.9%

        \[\leadsto 0.75 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{x}^{-3}}\right) \]

12. Simplified 1.9%

    \[\leadsto 0.75 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{-3}\right)} \]

13. Final simplification 1.9%

    \[\leadsto 0.75 \cdot \left({x}^{-3} \cdot {\pi}^{-0.5}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024109 
(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)))))))