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

Percentage Accurate: 100.0% → 100.0%

Time: 17.5s

Alternatives: 10

Speedup: 3.5×

• 99.6% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {x}^{-3} \cdot {x}^{-4}, \mathsf{fma}\left(0.75, \frac{{x}^{-4}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(1.875, Float64((x ^ -3.0) * (x ^ -4.0)), fma(0.75, Float64((x ^ -4.0) / abs(x)), fma(0.5, log(exp((x ^ -3.0))), Float64(1.0 / abs(x))))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

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. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

7. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)} - 1\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   2. expm1-log1p 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left({x}^{-2}\right)}^{2}}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   3. unpow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\left({x}^{-2} \cdot {x}^{-2}\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   4. pow-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{x}^{\left(2 \cdot -2\right)}}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   5. metadata-eval 100.0%

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

9. Simplified 100.0%

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

    1. expm1-log1p-u 100.0%

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

    2. expm1-udef 100.0%

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

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {x}^{-4}, \mathsf{fma}\left(0.75, \frac{\color{blue}{e^{\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)} - 1}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
12. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    2. expm1-log1p 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left({x}^{-2}\right)}^{2}}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    3. unpow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\left({x}^{-2} \cdot {x}^{-2}\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    4. pow-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{x}^{\left(2 \cdot -2\right)}}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    5. metadata-eval 100.0%

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

13. Simplified 100.0%

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

    1. expm1-log1p-u 100.0%

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

    2. expm1-udef 100.0%

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

    3. inv-pow 100.0%

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

    4. pow-pow 100.0%

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

    5. add-sqr-sqrt 100.0%

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

    6. fabs-sqr 100.0%

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

    7. add-sqr-sqrt 100.0%

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

    8. metadata-eval 100.0%

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

15. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

17. Simplified 100.0%

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

18. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-7}\right)} + -1\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (*
   (sqrt (/ 1.0 PI))
   (+
    (/ 0.5 (pow x 3.0))
    (+
     (/ 1.0 x)
     (fma 0.75 (pow x -5.0) (+ (exp (log1p (* 1.875 (pow x -7.0)))) -1.0)))))))

C

double code(double x) {
	return pow(exp(x), x) * (sqrt((1.0 / ((double) M_PI))) * ((0.5 / pow(x, 3.0)) + ((1.0 / x) + fma(0.75, pow(x, -5.0), (exp(log1p((1.875 * pow(x, -7.0)))) + -1.0)))));
}

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(Float64(1.0 / x) + fma(0.75, (x ^ -5.0), Float64(exp(log1p(Float64(1.875 * (x ^ -7.0)))) + -1.0))))))
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(1.875 * N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

6. Simplified 100.0%

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

   1. exp-prod 100.0%

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

8. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1.875}{{x}^{7}}\right)\right)}\right)\right)\right)\right) \]

   2. expm1-udef 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, \color{blue}{e^{\mathsf{log1p}\left(\frac{1.875}{{x}^{7}}\right)} - 1}\right)\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, e^{\mathsf{log1p}\left(\color{blue}{1.875 \cdot \frac{1}{{x}^{7}}}\right)} - 1\right)\right)\right)\right) \]

   4. pow-flip 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, e^{\mathsf{log1p}\left(1.875 \cdot \color{blue}{{x}^{\left(-7\right)}}\right)} - 1\right)\right)\right)\right) \]

   5. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{\color{blue}{-7}}\right)} - 1\right)\right)\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, \color{blue}{e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-7}\right)} - 1}\right)\right)\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \mathsf{fma}\left(0.75, {x}^{-5}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-7}\right)} + -1\right)\right)\right)\right) \]

Alternative 3: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.pow(Math.exp(x), x) * (Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (1.875 / Math.pow(x, 7.0))))));
}

Python

def code(x):
	return math.pow(math.exp(x), x) * (math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (1.875 / math.pow(x, 7.0))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) * (sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + ((1.0 / x) + ((0.75 / (x ^ 5.0)) + (1.875 / (x ^ 7.0))))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

6. Simplified 100.0%

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

   1. exp-prod 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in x around 0 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{5}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. associate-*r/ 100.0%

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

    4. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (1.875 / Math.pow(x, 7.0)))))) * Math.exp((x * x));
}

Python

def code(x):
	return (math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (1.875 / math.pow(x, 7.0)))))) * math.exp((x * x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + ((1.0 / x) + ((0.75 / (x ^ 5.0)) + (1.875 / (x ^ 7.0)))))) * exp((x * x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{5}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 5: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.exp((x * x)) * (Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((1.0 / x) + (0.75 / Math.pow(x, 5.0)))));
}

Python

def code(x):
	return math.exp((x * x)) * (math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((1.0 / x) + (0.75 / math.pow(x, 5.0)))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * (sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + ((1.0 / x) + (0.75 / (x ^ 5.0)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

6. Simplified 100.0%

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

7. Taylor expanded in x around inf 99.8%

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

8. Final simplification 99.8%

   \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right)\right)\right) \]

Alternative 6: 99.7% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

6. Simplified 100.0%

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

7. Taylor expanded in x around inf 99.8%

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

8. Taylor expanded in x around inf 99.8%

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

9. Final simplification 99.8%

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

Alternative 7: 99.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 8: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{{x}^{-6} \cdot 0.25}{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ (* (pow x -6.0) 0.25) PI)))

C

double code(double x) {
	return sqrt(((pow(x, -6.0) * 0.25) / ((double) M_PI)));
}

Java

public static double code(double x) {
	return Math.sqrt(((Math.pow(x, -6.0) * 0.25) / Math.PI));
}

Python

def code(x):
	return math.sqrt(((math.pow(x, -6.0) * 0.25) / math.pi))

Julia

function code(x)
	return sqrt(Float64(Float64((x ^ -6.0) * 0.25) / pi))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((((x ^ -6.0) * 0.25) / pi));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Power[x, -6.0], $MachinePrecision] * 0.25), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. *-commutative 34.4%

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

   3. associate-*r/ 34.4%

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

   4. metadata-eval 34.4%

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

   5. unpow2 34.4%

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

   6. unpow1 34.4%

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

   7. metadata-eval 34.4%

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

   8. pow-sqr 34.4%

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

   9. unpow1/2 34.4%

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

   10. unpow1/2 34.4%

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

   11. fabs-sqr 34.4%

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

   12. unpow1/2 34.4%

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

   13. unpow1/2 34.4%

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

   14. pow-sqr 34.4%

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

   15. metadata-eval 34.4%

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

   16. unpow1 34.4%

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

   17. unpow3 34.4%

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

6. Simplified 34.4%

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

7. Taylor expanded in x around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. *-commutative 1.8%

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

   3. associate-*l* 1.8%

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

   4. metadata-eval 1.8%

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

   5. *-lft-identity 1.8%

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

   6. cube-div 1.8%

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

   7. *-lft-identity 1.8%

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

   8. rem-exp-log 1.8%

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

   9. exp-neg 1.8%

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

   10. exp-prod 1.8%

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

   11. distribute-lft-neg-out 1.8%

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

   12. distribute-rgt-neg-in 1.8%

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

   13. metadata-eval 1.8%

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

   14. exp-to-pow 1.8%

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

   15. *-commutative 1.8%

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

9. Simplified 1.8%

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

    1. add-sqr-sqrt 1.8%

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

    2. sqrt-unprod 1.8%

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

    3. swap-sqr 1.8%

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

    4. add-sqr-sqrt 1.8%

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

    5. *-commutative 1.8%

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

    6. *-commutative 1.8%

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

    7. swap-sqr 1.8%

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

    8. pow-prod-up 1.8%

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

    9. metadata-eval 1.8%

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

    10. metadata-eval 1.8%

        \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{-6} \cdot \color{blue}{0.25}\right)} \]

11. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{-6} \cdot 0.25\right)}} \]
12. Step-by-step derivation

    1. associate-*l/ 1.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{-6} \cdot 0.25\right)}{\pi}}} \]

    2. *-lft-identity 1.8%

       \[\leadsto \sqrt{\frac{\color{blue}{{x}^{-6} \cdot 0.25}}{\pi}} \]

13. Simplified 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{{x}^{-6} \cdot 0.25}{\pi}}} \]

14. Final simplification 1.8%

    \[\leadsto \sqrt{\frac{{x}^{-6} \cdot 0.25}{\pi}} \]

Alternative 9: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{\pi} \cdot {x}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. *-commutative 34.4%

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

   3. associate-*r/ 34.4%

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

   4. metadata-eval 34.4%

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

   5. unpow2 34.4%

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

   6. unpow1 34.4%

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

   7. metadata-eval 34.4%

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

   8. pow-sqr 34.4%

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

   9. unpow1/2 34.4%

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

   10. unpow1/2 34.4%

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

   11. fabs-sqr 34.4%

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

   12. unpow1/2 34.4%

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

   13. unpow1/2 34.4%

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

   14. pow-sqr 34.4%

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

   15. metadata-eval 34.4%

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

   16. unpow1 34.4%

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

   17. unpow3 34.4%

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

6. Simplified 34.4%

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

7. Taylor expanded in x around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. *-commutative 1.8%

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

   3. associate-*l* 1.8%

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

   4. metadata-eval 1.8%

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

   5. *-lft-identity 1.8%

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

   6. cube-div 1.8%

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

   7. *-lft-identity 1.8%

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

   8. rem-exp-log 1.8%

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

   9. exp-neg 1.8%

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

   10. exp-prod 1.8%

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

   11. distribute-lft-neg-out 1.8%

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

   12. distribute-rgt-neg-in 1.8%

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

   13. metadata-eval 1.8%

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

   14. exp-to-pow 1.8%

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

   15. *-commutative 1.8%

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

9. Simplified 1.8%

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

    1. *-commutative 1.8%

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

    2. sqrt-div 1.8%

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

    3. metadata-eval 1.8%

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

    4. *-commutative 1.8%

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

    5. metadata-eval 1.8%

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

    6. pow-flip 1.8%

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

    7. associate-/r/ 1.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{0.5}}} \cdot \frac{1}{\sqrt{\pi}} \]

    8. frac-times 1.8%

       \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{{x}^{3}}{0.5} \cdot \sqrt{\pi}}} \]

    9. metadata-eval 1.8%

       \[\leadsto \frac{\color{blue}{1}}{\frac{{x}^{3}}{0.5} \cdot \sqrt{\pi}} \]

    10. div-inv 1.8%

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

    11. metadata-eval 1.8%

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

11. Applied egg-rr 1.8%

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

    1. associate-/r* 1.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{3} \cdot 2}}{\sqrt{\pi}}} \]

    2. *-commutative 1.8%

       \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot {x}^{3}}}}{\sqrt{\pi}} \]

    3. associate-/r* 1.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{3}}}}{\sqrt{\pi}} \]

    4. metadata-eval 1.8%

       \[\leadsto \frac{\frac{\color{blue}{0.5}}{{x}^{3}}}{\sqrt{\pi}} \]

    5. associate-/r* 1.8%

       \[\leadsto \color{blue}{\frac{0.5}{{x}^{3} \cdot \sqrt{\pi}}} \]

13. Simplified 1.8%

    \[\leadsto \color{blue}{\frac{0.5}{{x}^{3} \cdot \sqrt{\pi}}} \]

14. Final simplification 1.8%

    \[\leadsto \frac{0.5}{\sqrt{\pi} \cdot {x}^{3}} \]

Alternative 10: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-3} \cdot 0.5}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. *-commutative 34.4%

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

   3. associate-*r/ 34.4%

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

   4. metadata-eval 34.4%

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

   5. unpow2 34.4%

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

   6. unpow1 34.4%

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

   7. metadata-eval 34.4%

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

   8. pow-sqr 34.4%

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

   9. unpow1/2 34.4%

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

   10. unpow1/2 34.4%

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

   11. fabs-sqr 34.4%

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

   12. unpow1/2 34.4%

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

   13. unpow1/2 34.4%

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

   14. pow-sqr 34.4%

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

   15. metadata-eval 34.4%

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

   16. unpow1 34.4%

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

   17. unpow3 34.4%

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

6. Simplified 34.4%

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

7. Taylor expanded in x around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. *-commutative 1.8%

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

   3. associate-*l* 1.8%

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

   4. metadata-eval 1.8%

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

   5. *-lft-identity 1.8%

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

   6. cube-div 1.8%

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

   7. *-lft-identity 1.8%

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

   8. rem-exp-log 1.8%

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

   9. exp-neg 1.8%

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

   10. exp-prod 1.8%

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

   11. distribute-lft-neg-out 1.8%

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

   12. distribute-rgt-neg-in 1.8%

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

   13. metadata-eval 1.8%

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

   14. exp-to-pow 1.8%

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

   15. *-commutative 1.8%

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

9. Simplified 1.8%

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

    1. *-commutative 1.8%

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

    2. sqrt-div 1.8%

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

    3. metadata-eval 1.8%

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

    4. un-div-inv 1.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-3}}{\sqrt{\pi}}} \]

11. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-3}}{\sqrt{\pi}}} \]

12. Final simplification 1.8%

    \[\leadsto \frac{{x}^{-3} \cdot 0.5}{\sqrt{\pi}} \]

Reproduce

herbie shell --seed 2023318 
(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)))))))