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

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 11

Speedup: 4.0×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, t\_0 \cdot \left(1 + \mathsf{fma}\left(0.75, {t\_0}^{4}, 1.875 \cdot \log \left(e^{{x}^{-6}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (/ (pow (exp x) x) (sqrt PI))
    (fma
     0.5
     (/ 1.0 (pow (fabs x) 3.0))
     (*
      t_0
      (+ 1.0 (fma 0.75 (pow t_0 4.0) (* 1.875 (log (exp (pow x -6.0)))))))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(0.5, (1.0 / pow(fabs(x), 3.0)), (t_0 * (1.0 + fma(0.75, pow(t_0, 4.0), (1.875 * log(exp(pow(x, -6.0))))))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(0.5, Float64(1.0 / (abs(x) ^ 3.0)), Float64(t_0 * Float64(1.0 + fma(0.75, (t_0 ^ 4.0), Float64(1.875 * log(exp((x ^ -6.0)))))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(0.75 * N[Power[t$95$0, 4.0], $MachinePrecision] + N[(1.875 * N[Log[N[Exp[N[Power[x, -6.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. fma-undefine 100.0%

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

   2. pow-flip 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l/ 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) / x);
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) / x);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around 0 19.1%

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

   1. +-commutative 19.1%

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

9. Simplified 19.1%

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

10. Taylor expanded in x around inf 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
11. Step-by-step derivation

    1. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

    2. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

    3. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

12. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
13. Step-by-step derivation

    1. unpow2 100.0%

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

14. Applied egg-rr 100.0%

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

Alternative 4: 99.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

9. Simplified 99.0%

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

    1. unpow2 100.0%

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

11. Applied egg-rr 99.0%

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

    1. add-sqr-sqrt 99.0%

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

    2. unpow-prod-down 99.0%

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

13. Applied egg-rr 99.0%

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

    1. pow-sqr 99.0%

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

    2. *-commutative 99.0%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 5: 99.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

9. Simplified 99.0%

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

    1. unpow2 100.0%

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

11. Applied egg-rr 99.0%

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

    1. add-exp-log 99.0%

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

    2. log-div 99.0%

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

    3. pow-exp 99.0%

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

    4. add-log-exp 99.0%

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

    5. pow2 99.0%

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

13. Applied egg-rr 99.0%

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

14. Final simplification 99.0%

    \[\leadsto \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)}{x} \cdot e^{{x}^{2} - \log \left(\sqrt{\pi}\right)} \]
15. Add Preprocessing

Alternative 6: 99.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

9. Simplified 99.0%

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

    1. unpow2 100.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 7: 99.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

9. Simplified 99.0%

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

    1. unpow2 100.0%

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

11. Applied egg-rr 99.0%

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

    1. *-un-lft-identity 99.0%

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

    2. pow-exp 99.0%

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

    3. pow2 99.0%

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

13. Applied egg-rr 99.0%

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

    1. *-lft-identity 99.0%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 8: 99.8% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

9. Simplified 98.9%

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

    1. unpow2 100.0%

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

11. Applied egg-rr 98.9%

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

Alternative 9: 99.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 98.8%

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

   1. associate-*l/ 98.8%

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

   2. unpow2 98.8%

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

   3. pow-exp 98.8%

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

   4. sqrt-div 98.8%

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

   5. metadata-eval 98.8%

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

   6. div-inv 98.8%

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

   7. pow-exp 98.8%

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

   8. unpow2 98.8%

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

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{x}} \]
10. Step-by-step derivation

    1. associate-/l/ 98.8%

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

11. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}}} \]
12. Step-by-step derivation

    1. unpow2 98.8%

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

    2. pow-exp 98.8%

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

13. Applied egg-rr 98.8%

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

Alternative 10: 99.7% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 98.8%

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

   1. associate-*l/ 98.8%

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

   2. unpow2 98.8%

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

   3. pow-exp 98.8%

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

   4. sqrt-div 98.8%

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

   5. metadata-eval 98.8%

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

   6. div-inv 98.8%

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

   7. pow-exp 98.8%

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

   8. unpow2 98.8%

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

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{x}} \]
10. Step-by-step derivation

    1. associate-/l/ 98.8%

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

11. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}}} \]
12. Step-by-step derivation

    1. unpow2 100.0%

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

13. Applied egg-rr 98.8%

    \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{x \cdot \sqrt{\pi}} \]
14. Add Preprocessing

Alternative 11: 2.3% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right)}^{2}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}\right)}^{2}} \]

7. Taylor expanded in x around inf 98.8%

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

8. Taylor expanded in x around 0 2.4%

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

   1. associate-*l/ 2.4%

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

   2. *-lft-identity 2.4%

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

   3. unpow-1 2.4%

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

   4. metadata-eval 2.4%

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

   5. pow-sqr 2.4%

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

   6. rem-sqrt-square 2.4%

      \[\leadsto \frac{\color{blue}{\left|{\pi}^{-0.5}\right|}}{x} \]

   7. rem-square-sqrt 2.4%

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

   8. fabs-sqr 2.4%

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

   9. rem-square-sqrt 2.4%

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

10. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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)))))))