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

Percentage Accurate: 100.0% → 100.0%

Time: 15.9s

Alternatives: 12

Speedup: 3.3×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[Power[N[Power[N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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%

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

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{\sqrt[3]{{\pi}^{1.5}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (/
   (fma
    0.5
    (pow x -3.0)
    (/ (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (/ 1.875 (pow x 6.0)))) x))
   (cbrt (pow PI 1.5)))))

C

double code(double x) {
	return pow(exp(x), x) * (fma(0.5, pow(x, -3.0), ((1.0 + ((0.75 / pow(x, 4.0)) + (1.875 / pow(x, 6.0)))) / x)) / cbrt(pow(((double) M_PI), 1.5)));
}

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(fma(0.5, (x ^ -3.0), Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(1.875 / (x ^ 6.0)))) / x)) / cbrt((pi ^ 1.5))))
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $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%

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

   1. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   4. pow1 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   5. pow1/2 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \]
11. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \]

12. Simplified 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \]

13. Taylor expanded in x around 0 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{4}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}}{x}\right)}{\sqrt[3]{{\pi}^{1.5}}} \]
14. Step-by-step derivation

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. associate-*r/ 100.0%

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

    4. metadata-eval 100.0%

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

15. Simplified 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}}{x}\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

16. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{\sqrt[3]{{\pi}^{1.5}}} \]
17. Add Preprocessing

Alternative 3: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ {\left(e^{x}\right)}^{x} \cdot \left(t\_0 \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + t\_0 \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (*
    (pow (exp x) x)
    (+
     (* t_0 (+ (/ 1.875 (pow x 7.0)) (/ 0.75 (pow x 5.0))))
     (* t_0 (+ (/ 0.5 (pow x 3.0)) (/ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	return pow(exp(x), x) * ((t_0 * ((1.875 / pow(x, 7.0)) + (0.75 / pow(x, 5.0)))) + (t_0 * ((0.5 / pow(x, 3.0)) + (1.0 / x))));
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return Math.pow(Math.exp(x), x) * ((t_0 * ((1.875 / Math.pow(x, 7.0)) + (0.75 / Math.pow(x, 5.0)))) + (t_0 * ((0.5 / Math.pow(x, 3.0)) + (1.0 / x))));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	return math.pow(math.exp(x), x) * ((t_0 * ((1.875 / math.pow(x, 7.0)) + (0.75 / math.pow(x, 5.0)))) + (t_0 * ((0.5 / math.pow(x, 3.0)) + (1.0 / x))))

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	return Float64((exp(x) ^ x) * Float64(Float64(t_0 * Float64(Float64(1.875 / (x ^ 7.0)) + Float64(0.75 / (x ^ 5.0)))) + Float64(t_0 * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = (exp(x) ^ x) * ((t_0 * ((1.875 / (x ^ 7.0)) + (0.75 / (x ^ 5.0)))) + (t_0 * ((0.5 / (x ^ 3.0)) + (1.0 / x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(t$95$0 * N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / 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%

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

   1. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. associate-*r/ 100.0%

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

    4. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 99.6% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

10. Taylor expanded in x around inf 99.8%

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

11. Final simplification 99.8%

    \[\leadsto e^{{x}^{2}} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right)}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 7: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

10. Taylor expanded in x around inf 99.8%

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

11. Taylor expanded in x around inf 99.8%

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

    1. associate-*r* 99.8%

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

    2. associate-*r/ 99.8%

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

    3. metadata-eval 99.8%

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

    4. distribute-rgt-out 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

    \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot e^{{x}^{2}} \]
15. Add Preprocessing

Alternative 8: 99.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 99.7%

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

   1. associate-+r+ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+l+ 99.7%

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

   4. metadata-eval 99.7%

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

   5. fabs-div 99.7%

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

   6. unpow1 99.7%

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

   7. sqr-pow 99.7%

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

   8. fabs-sqr 99.7%

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

   9. sqr-pow 99.7%

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

   10. unpow1 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*r* 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \frac{1}{x}\right)\right) \]
12. Add Preprocessing

Alternative 9: 99.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 99.7%

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

   1. associate-+r+ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+l+ 99.7%

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

   4. metadata-eval 99.7%

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

   5. fabs-div 99.7%

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

   6. unpow1 99.7%

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

   7. sqr-pow 99.7%

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

   8. fabs-sqr 99.7%

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

   9. sqr-pow 99.7%

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

   10. unpow1 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*r* 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in x around inf 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

13. Simplified 99.7%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{x}} \]
14. Step-by-step derivation

    1. clear-num 99.7%

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

    2. inv-pow 99.7%

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

    3. sqrt-div 99.7%

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

    4. metadata-eval 99.7%

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

    5. pow1/2 99.7%

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

    6. metadata-eval 99.7%

       \[\leadsto e^{x \cdot x} \cdot {\left(\frac{x}{\frac{1}{{\pi}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}}\right)}^{-1} \]

    7. pow-pow 99.7%

       \[\leadsto e^{x \cdot x} \cdot {\left(\frac{x}{\frac{1}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}}}\right)}^{-1} \]

    8. pow1/3 99.7%

       \[\leadsto e^{x \cdot x} \cdot {\left(\frac{x}{\frac{1}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}}}\right)}^{-1} \]

    9. associate-/r/ 99.7%

       \[\leadsto e^{x \cdot x} \cdot {\color{blue}{\left(\frac{x}{1} \cdot \sqrt[3]{{\pi}^{1.5}}\right)}}^{-1} \]

    10. /-rgt-identity 99.7%

        \[\leadsto e^{x \cdot x} \cdot {\left(\color{blue}{x} \cdot \sqrt[3]{{\pi}^{1.5}}\right)}^{-1} \]

    11. pow1/3 99.7%

        \[\leadsto e^{x \cdot x} \cdot {\left(x \cdot \color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}\right)}^{-1} \]

    12. pow-pow 99.7%

        \[\leadsto e^{x \cdot x} \cdot {\left(x \cdot \color{blue}{{\pi}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{-1} \]

    13. metadata-eval 99.7%

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

    14. pow1/2 99.7%

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

15. Applied egg-rr 99.7%

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

16. Final simplification 99.7%

    \[\leadsto e^{x \cdot x} \cdot {\left(x \cdot \sqrt{\pi}\right)}^{-1} \]
17. Add Preprocessing

Alternative 10: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 99.7%

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

   1. associate-+r+ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+l+ 99.7%

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

   4. metadata-eval 99.7%

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

   5. fabs-div 99.7%

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

   6. unpow1 99.7%

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

   7. sqr-pow 99.7%

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

   8. fabs-sqr 99.7%

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

   9. sqr-pow 99.7%

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

   10. unpow1 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*r* 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in x around inf 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

13. Simplified 99.7%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{x}} \]
14. Step-by-step derivation

    1. *-un-lft-identity 2.3%

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

    2. div-inv 2.3%

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

    3. div-inv 2.3%

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

    4. inv-pow 2.3%

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

    5. sqrt-pow1 2.3%

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

    6. metadata-eval 2.3%

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

15. Applied egg-rr 99.7%

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

    1. *-lft-identity 2.3%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

    \[\leadsto e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \]
19. Add Preprocessing

Alternative 11: 2.3% accurate, 19.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 40.7%

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

   1. associate-*r* 40.7%

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

   2. *-commutative 40.7%

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

   3. *-commutative 40.7%

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

   4. associate-/r* 41.9%

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

   5. metadata-eval 41.9%

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

   6. fabs-div 41.9%

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

   7. unpow1 41.9%

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

   8. sqr-pow 41.9%

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

   9. fabs-sqr 41.9%

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

   10. sqr-pow 41.9%

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

   11. unpow1 41.9%

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

   12. associate-/r* 40.7%

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

   13. unpow2 40.7%

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

   14. cube-mult 40.7%

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

   15. exp-to-pow 40.7%

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

   16. *-commutative 40.7%

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

   17. exp-neg 41.9%

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

   18. distribute-lft-neg-in 41.9%

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

   19. metadata-eval 41.9%

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

   20. *-commutative 41.9%

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

   21. exp-to-pow 41.9%

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

7. Simplified 41.9%

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

8. Taylor expanded in x around 0 2.3%

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

   1. +-commutative 2.3%

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

   2. associate-*r* 2.3%

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

   3. associate-*r* 2.3%

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

   4. distribute-rgt-out 2.3%

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

   5. associate-*r/ 2.3%

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

   6. metadata-eval 2.3%

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

   7. associate-*r/ 2.3%

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

   8. metadata-eval 2.3%

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

10. Simplified 2.3%

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

11. Taylor expanded in x around inf 2.3%

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

    1. associate-*l/ 2.3%

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

    2. *-lft-identity 2.3%

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

13. Simplified 2.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{x}} \]
14. Step-by-step derivation

    1. *-un-lft-identity 2.3%

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

    2. div-inv 2.3%

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

    3. div-inv 2.3%

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

    4. inv-pow 2.3%

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

    5. sqrt-pow1 2.3%

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

    6. metadata-eval 2.3%

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

15. Applied egg-rr 2.3%

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

    1. *-lft-identity 2.3%

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

17. Simplified 2.3%

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

18. Final simplification 2.3%

    \[\leadsto 0.5 \cdot \frac{{\pi}^{-0.5}}{x} \]
19. Add Preprocessing

Alternative 12: 2.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 99.7%

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

   1. associate-+r+ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+l+ 99.7%

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

   4. metadata-eval 99.7%

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

   5. fabs-div 99.7%

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

   6. unpow1 99.7%

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

   7. sqr-pow 99.7%

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

   8. fabs-sqr 99.7%

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

   9. sqr-pow 99.7%

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

   10. unpow1 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*r* 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in x around inf 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

13. Simplified 99.7%

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

14. Taylor expanded in x around 0 2.3%

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

15. Final simplification 2.3%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{x} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))))