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

Percentage Accurate: 100.0% → 100.0%

Time: 32.5s

Alternatives: 8

Speedup: 2.9×

• 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, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (pow (pow PI 0.25) 2.0))
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	return (pow(exp(x), x) / pow(pow(((double) M_PI), 0.25), 2.0)) * fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / ((pi ^ 0.25) ^ 2.0)) * fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. exp-prod 100.0%

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

10. Applied egg-rr 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. pow2 100.0%

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

    3. pow1/2 100.0%

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

    4. sqrt-pow1 100.0%

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

    5. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.25}}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
13. Step-by-step derivation

    1. *-un-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. inv-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
15. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

16. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
17. Add Preprocessing

Alternative 2: 100.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   0.5
   (pow (/ 1.0 x) 3.0)
   (*
    (/ 1.0 (fabs x))
    (+ 1.0 (+ (* 1.875 (pow x -6.0)) (* 0.75 (pow x -4.0))))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(0.5, pow((1.0 / x), 3.0), ((1.0 / fabs(x)) * (1.0 + ((1.875 * pow(x, -6.0)) + (0.75 * pow(x, -4.0))))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(0.5, (Float64(1.0 / x) ^ 3.0), Float64(Float64(1.0 / abs(x)) * Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(0.75 * (x ^ -4.0)))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Power[N[(1.0 / x), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $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%

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

5. Taylor expanded in x around 0 100.0%

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

   1. rem-square-sqrt 100.0%

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

   2. fabs-sqr 100.0%

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

   3. rem-square-sqrt 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow-sqr 100.0%

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

   6. associate-/r* 100.0%

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

   7. exp-to-pow 100.0%

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

   8. *-commutative 100.0%

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

   9. rec-exp 100.0%

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

   10. mul-1-neg 100.0%

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

   11. associate-*r* 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

   14. exp-to-pow 100.0%

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

   15. exp-to-pow 100.0%

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

   16. *-commutative 100.0%

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

   17. exp-to-pow 100.0%

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

   18. *-commutative 100.0%

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

   19. div-exp 100.0%

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

   20. unsub-neg 100.0%

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

   21. distribute-lft-neg-in 100.0%

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

7. Simplified 100.0%

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

   1. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. inv-pow 100.0%

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

   5. pow-pow 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. fabs-sqr 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. metadata-eval 100.0%

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

    2. cube-div 100.0%

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

    3. rem-cube-cbrt 100.0%

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

    4. rem-cbrt-cube 100.0%

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

    5. add-sqr-sqrt 100.0%

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

    6. fabs-sqr 100.0%

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

    7. add-sqr-sqrt 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-undefine 100.0%

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

   3. rem-exp-log 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-undefine 100.0%

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

   3. rem-exp-log 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.1%

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

14. Final simplification 99.1%

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

Alternative 6: 99.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (pow x 2.0)) (sqrt PI))
  (fma 0.75 (pow x -5.0) (fma 1.875 (pow x -7.0) (/ 1.0 x)))))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / sqrt(((double) M_PI))) * fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), (1.0 / x)));
}

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) * fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(1.0 / x))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $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%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.1%

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

    1. fma-undefine 99.1%

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

    2. distribute-lft-in 20.6%

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

    3. pow2 20.6%

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

    4. pow2 20.6%

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

    5. add-sqr-sqrt 20.6%

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

    6. fabs-sqr 20.6%

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

    7. add-sqr-sqrt 20.6%

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

15. Applied egg-rr 20.6%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5}\right) + \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x}\right)} \]
16. Step-by-step derivation

    1. distribute-lft-out 99.1%

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

    2. fma-undefine 99.1%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x}\right)\right)} \]

17. Simplified 99.1%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x}\right)\right)} \]
18. Add Preprocessing

Alternative 7: 99.6% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = exp((x ^ 2.0)) * (sqrt((1.0 / pi)) / abs(x));
end

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[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%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.1%

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

14. Taylor expanded in x around inf 99.1%

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

    1. associate-*r/ 99.1%

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

    2. *-commutative 99.1%

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

    3. associate-/l* 99.1%

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

16. Simplified 99.1%

    \[\leadsto \color{blue}{e^{{x}^{2}} \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}} \]
17. Add Preprocessing

Alternative 8: 2.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (sqrt (/ 1.0 PI)) (fabs x)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[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%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 2.4%

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

14. Taylor expanded in x around inf 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}} \]
15. Step-by-step derivation

    1. associate-*r/ 2.4%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}} \cdot 1}{\left|x\right|}} \]

    2. *-rgt-identity 2.4%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{\left|x\right|} \]

16. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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)))))))