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

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 7

Speedup: 3.4×

• 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, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

9. Taylor expanded in x around 0 99.4%

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

10. Final simplification 99.4%

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

Alternative 3: 99.6% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 100.0%

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

9. Taylor expanded in x around inf 99.4%

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

    1. associate-*r* 99.4%

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

    2. distribute-rgt-out 99.4%

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

    3. unpow-1 99.4%

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

    4. metadata-eval 99.4%

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

    5. pow-sqr 99.4%

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

    6. rem-sqrt-square 99.4%

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

    7. sqr-pow 99.4%

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

    8. fabs-sqr 99.4%

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

    9. sqr-pow 99.4%

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

    10. associate-*r/ 99.4%

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

    11. metadata-eval 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64((exp(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[Power[N[Exp[x], $MachinePrecision], x], $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%

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 100.0%

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

9. Taylor expanded in x around inf 99.4%

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

    1. unpow-1 99.4%

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

    2. metadata-eval 99.4%

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

    3. pow-sqr 99.4%

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

    4. rem-sqrt-square 99.4%

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

    5. sqr-pow 99.4%

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

    6. fabs-sqr 99.4%

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

    7. sqr-pow 99.4%

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

    8. associate-*l/ 99.4%

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

    9. *-lft-identity 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{{\pi}^{-0.5}}{x} \]
13. Add Preprocessing

Alternative 5: 51.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5}}{x} \cdot \mathsf{fma}\left(x, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ (pow PI -0.5) x) (fma x x 1.0)))

C

double code(double x) {
	return (pow(((double) M_PI), -0.5) / x) * fma(x, x, 1.0);
}

Julia

function code(x)
	return Float64(Float64((pi ^ -0.5) / x) * fma(x, x, 1.0))
end

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 58.0%

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

    1. +-commutative 58.0%

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

    2. unpow2 58.0%

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

    3. fma-def 58.0%

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

13. Simplified 58.0%

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

    1. expm1-log1p-u 58.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{x}\right)\right)} \]

    2. expm1-udef 58.0%

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

    3. inv-pow 58.0%

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

    4. sqrt-pow1 58.0%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right) \cdot \frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right)} - 1 \]

    5. metadata-eval 58.0%

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

15. Applied egg-rr 58.0%

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

    1. expm1-def 58.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right) \cdot \frac{{\pi}^{-0.5}}{x}\right)\right)} \]

    2. expm1-log1p 58.0%

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

    3. *-commutative 58.0%

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

17. Simplified 58.0%

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

18. Final simplification 58.0%

    \[\leadsto \frac{{\pi}^{-0.5}}{x} \cdot \mathsf{fma}\left(x, x, 1\right) \]
19. Add Preprocessing

Alternative 6: 5.4% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 5.6%

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

    1. distribute-rgt-out 5.6%

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

    2. unpow-1 5.6%

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

    3. metadata-eval 5.6%

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

    4. pow-sqr 5.6%

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

    5. rem-sqrt-square 5.6%

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

    6. sqr-pow 5.6%

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

    7. fabs-sqr 5.6%

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

    8. sqr-pow 5.6%

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

13. Simplified 5.6%

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

14. Final simplification 5.6%

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

Alternative 7: 5.4% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l+ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 58.0%

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

    1. +-commutative 58.0%

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

    2. unpow2 58.0%

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

    3. fma-def 58.0%

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

13. Simplified 58.0%

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

14. Taylor expanded in x around inf 5.6%

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

    1. unpow-1 5.6%

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

    2. metadata-eval 5.6%

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

    3. pow-sqr 5.6%

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

    4. rem-sqrt-square 5.6%

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

    5. metadata-eval 5.6%

       \[\leadsto x \cdot \left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

    6. pow-sqr 5.6%

       \[\leadsto x \cdot \left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \]

    7. fabs-sqr 5.6%

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

    8. pow-sqr 5.6%

       \[\leadsto x \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \]

    9. metadata-eval 5.6%

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

16. Simplified 5.6%

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

17. Final simplification 5.6%

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

Reproduce

herbie shell --seed 2024011 
(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)))))))