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

Percentage Accurate: 100.0% → 100.0%

Time: 20.8s

Alternatives: 13

Speedup: 3.3×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(e^{{x}^{2}}\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1}{x}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (expm1 (log1p (exp (pow x 2.0))))
  (*
   (sqrt (/ 1.0 PI))
   (+
    (/ 0.5 (pow x 3.0))
    (+ (/ 1.875 (pow x 7.0)) (fma 0.75 (pow x -5.0) (/ 1.0 x)))))))

C

double code(double x) {
	return expm1(log1p(exp(pow(x, 2.0)))) * (sqrt((1.0 / ((double) M_PI))) * ((0.5 / pow(x, 3.0)) + ((1.875 / pow(x, 7.0)) + fma(0.75, pow(x, -5.0), (1.0 / x)))));
}

Julia

function code(x)
	return Float64(expm1(log1p(exp((x ^ 2.0)))) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(Float64(1.875 / (x ^ 7.0)) + fma(0.75, (x ^ -5.0), Float64(1.0 / x))))))
end

Wolfram

code[x_] := N[(N[(Exp[N[Log[1 + N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. pow2 100.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{{x}^{2}}}\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1}{x}\right)\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{{x}^{2}}\right)\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1}{x}\right)\right)\right)\right) \]

10. Final simplification 100.0%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{{x}^{2}}\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{1.875}{{x}^{7}} + \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1}{x}\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. exp-prod 100.0%

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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)) + ((0.5 / Math.pow(x, 3.0)) + ((1.875 / Math.pow(x, 7.0)) + (1.0 / x)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. exp-prod 100.0%

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

    1. associate-+r+ 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. associate-+l+ 100.0%

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

    5. associate-*r/ 100.0%

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

    6. metadata-eval 100.0%

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

    7. associate-*r/ 100.0%

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

    8. metadata-eval 100.0%

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

    9. associate-*r/ 100.0%

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

    10. metadata-eval 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 5: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-+r+ 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. metadata-eval 99.4%

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

   6. cube-div 99.4%

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

   7. associate-*r* 99.4%

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

   8. associate-*r* 99.4%

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

   9. distribute-rgt-out 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 6: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

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)) + ((0.5 / Math.pow(x, 3.0)) + (1.0 / x))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. exp-prod 100.0%

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in x around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. associate-+l+ 99.4%

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

    3. associate-*r/ 99.4%

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

    4. metadata-eval 99.4%

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

    5. +-commutative 99.4%

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

    6. associate-*r/ 99.4%

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

    7. metadata-eval 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 7: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. metadata-eval 99.4%

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

   2. cube-div 99.4%

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

   3. associate-*r* 99.4%

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

   4. distribute-rgt-out 99.4%

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

   5. cube-div 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-*r/ 99.4%

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

   8. metadata-eval 99.4%

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

10. Simplified 99.4%

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

    1. add-sqr-sqrt 99.4%

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

    2. pow2 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 8: 99.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. metadata-eval 99.4%

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

   2. cube-div 99.4%

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

   3. associate-*r* 99.4%

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

   4. distribute-rgt-out 99.4%

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

   5. cube-div 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-*r/ 99.4%

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

   8. metadata-eval 99.4%

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

10. Simplified 99.4%

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

    1. expm1-log1p-u 99.4%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

    2. expm1-udef 4.9%

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

    3. *-commutative 4.9%

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

    4. sqrt-div 4.9%

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

    5. metadata-eval 4.9%

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

    6. un-div-inv 4.9%

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

    7. div-inv 4.9%

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

    8. fma-def 4.9%

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

    9. pow-flip 4.9%

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

    10. metadata-eval 4.9%

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

12. Applied egg-rr 4.9%

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

    1. expm1-def 99.4%

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

    2. expm1-log1p 99.4%

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

14. Simplified 99.4%

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

    1. fma-udef 99.4%

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

16. Applied egg-rr 99.4%

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

17. Final simplification 99.4%

    \[\leadsto e^{x \cdot x} \cdot \frac{\frac{1}{x} + 0.5 \cdot {x}^{-3}}{\sqrt{\pi}} \]
18. Add Preprocessing

Alternative 9: 99.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

    1. expm1-log1p-u 99.4%

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

    2. expm1-udef 99.4%

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

    3. clear-num 99.4%

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

    4. un-div-inv 99.4%

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

    5. pow2 99.4%

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

    6. sqrt-div 99.4%

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

    7. metadata-eval 99.4%

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

    8. associate-/r/ 99.4%

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

    9. /-rgt-identity 99.4%

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

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 99.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}}\right)\right)} \]

    2. expm1-log1p 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

    \[\leadsto \frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}} \]
16. Add Preprocessing

Alternative 10: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

    \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}}}{x} \]
12. Add Preprocessing

Alternative 11: 5.4% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 5.5%

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

    1. distribute-rgt-out 5.5%

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

13. Simplified 5.5%

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

    1. expm1-log1p-u 5.5%

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

    2. expm1-udef 5.5%

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

    3. *-commutative 5.5%

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

    4. sqrt-div 5.5%

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

    5. metadata-eval 5.5%

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

    6. un-div-inv 5.5%

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

15. Applied egg-rr 5.5%

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

    1. expm1-def 5.5%

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

    2. expm1-log1p 5.5%

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

17. Simplified 5.5%

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

18. Final simplification 5.5%

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

Alternative 12: 5.4% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 5.5%

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

    1. distribute-rgt-out 5.5%

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

13. Simplified 5.5%

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

14. Taylor expanded in x around inf 5.5%

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

15. Final simplification 5.5%

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

Alternative 13: 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 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 2.3%

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

    1. associate-*l/ 2.3%

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

    2. *-lft-identity 2.3%

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

13. Simplified 2.3%

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

    1. expm1-log1p-u 2.3%

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

    2. expm1-udef 1.7%

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

    3. inv-pow 1.7%

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

    4. sqrt-pow1 1.7%

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

    5. metadata-eval 1.7%

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

15. Applied egg-rr 1.7%

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

    1. expm1-def 2.3%

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

    2. expm1-log1p 2.3%

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

17. Simplified 2.3%

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

18. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))