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

Percentage Accurate: 100.0% → 100.0%

Time: 16.1s

Alternatives: 11

Speedup: 4.2×

• 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, 3.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.pow(Math.pow(Math.PI, 0.25), 2.0)) * (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (((1.0 + Math.pow(x, -5.0)) + -1.0) * (0.75 + (1.875 / (x * x)))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.pow(math.pow(math.pi, 0.25), 2.0)) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (((1.0 + math.pow(x, -5.0)) + -1.0) * (0.75 + (1.875 / (x * x)))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / ((pi ^ 0.25) ^ 2.0)) * (((1.0 + (0.5 / (x * x))) / abs(x)) + (((1.0 + (x ^ -5.0)) + -1.0) * (0.75 + (1.875 / (x * x)))));
end

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-udef 100.0%

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

   3. add-exp-log 100.0%

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

   4. +-commutative 100.0%

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

   5. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

8. Taylor expanded in x around inf 100.0%

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

   1. unpow2 100.0%

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

   2. exp-prod 100.0%

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

10. Simplified 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. pow2 100.0%

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

    3. pow1/2 100.0%

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

    4. sqrt-pow1 100.0%

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

    5. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-udef 100.0%

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

   3. add-exp-log 100.0%

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

   4. +-commutative 100.0%

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

   5. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

8. Taylor expanded in x around inf 100.0%

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

   1. unpow2 100.0%

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

   2. exp-prod 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. pow-flip 100.0%

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

   4. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 5: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 52.3% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-udef 100.0%

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

   3. add-exp-log 100.0%

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

   4. +-commutative 100.0%

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

   5. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

8. Taylor expanded in x around 0 43.1%

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

   1. unpow2 43.1%

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

10. Simplified 43.1%

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

11. Final simplification 43.1%

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

Alternative 7: 52.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 98.9%

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

9. Taylor expanded in x around 0 43.1%

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

    1. unpow2 43.1%

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

11. Simplified 43.1%

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

    1. clear-num 43.1%

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

    2. associate-/r/ 43.1%

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

    3. pow-flip 43.1%

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

    4. metadata-eval 43.1%

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

13. Applied egg-rr 43.1%

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

14. Final simplification 43.1%

    \[\leadsto \frac{1 + x \cdot x}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + 1.875 \cdot {x}^{-7}\right) \]

Alternative 8: 52.3% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 98.9%

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

9. Taylor expanded in x around 0 43.1%

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

    1. unpow2 43.1%

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

11. Simplified 43.1%

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

12. Taylor expanded in x around inf 43.1%

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

    1. *-commutative 43.1%

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

    2. unpow2 43.1%

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

14. Simplified 43.1%

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

15. Final simplification 43.1%

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

Alternative 9: 5.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{x}{x}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (/ 1.5 x) (/ x (/ x x)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((1.5 / x) + (x / (x / x)));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * ((1.5 / x) + (x / (x / x)));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((1.5 / x) + (x / (x / x)))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(1.5 / x) + Float64(x / Float64(x / x))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((1.5 / x) + (x / (x / x)));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5 / x), $MachinePrecision] + N[(x / N[(x / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 98.9%

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

9. Taylor expanded in x around 0 43.1%

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

    1. unpow2 43.1%

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

11. Simplified 43.1%

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

12. Taylor expanded in x around inf 43.1%

    \[\leadsto \color{blue}{1.5 \cdot \left(\frac{1}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
13. Step-by-step derivation

    1. associate-*r* 43.1%

       \[\leadsto \color{blue}{\left(1.5 \cdot \frac{1}{\left|x\right|}\right) \cdot \sqrt{\frac{1}{\pi}}} + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}} \]

    2. distribute-rgt-out 43.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(1.5 \cdot \frac{1}{\left|x\right|} + \frac{{x}^{2}}{\left|x\right|}\right)} \]

    3. associate-*r/ 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{1.5 \cdot 1}{\left|x\right|}} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    4. metadata-eval 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{1.5}}{\left|x\right|} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    5. unpow1 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{\left|\color{blue}{{x}^{1}}\right|} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    6. sqr-pow 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    7. fabs-sqr 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    8. sqr-pow 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{\color{blue}{{x}^{1}}} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    9. unpow1 43.1%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{\color{blue}{x}} + \frac{{x}^{2}}{\left|x\right|}\right) \]

    10. unpow2 43.1%

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{\color{blue}{x \cdot x}}{\left|x\right|}\right) \]

    11. associate-/l* 5.1%

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \color{blue}{\frac{x}{\frac{\left|x\right|}{x}}}\right) \]

    12. unpow1 5.1%

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

    13. sqr-pow 5.1%

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{x}}\right) \]

    14. fabs-sqr 5.1%

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{x}}\right) \]

    15. sqr-pow 5.1%

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

    16. unpow1 5.1%

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{\color{blue}{x}}{x}}\right) \]

14. Simplified 5.1%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{x}{x}}\right)} \]

15. Final simplification 5.1%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.5}{x} + \frac{x}{\frac{x}{x}}\right) \]

Alternative 10: 5.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 98.9%

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

9. Taylor expanded in x around 0 43.1%

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

    1. unpow2 43.1%

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

11. Simplified 43.1%

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

12. Taylor expanded in x around inf 43.1%

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

    1. *-commutative 43.1%

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

    2. unpow2 43.1%

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

    3. associate-/l* 5.1%

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

    4. unpow1 5.1%

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

    5. sqr-pow 5.1%

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

    6. fabs-sqr 5.1%

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

    7. sqr-pow 5.1%

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

    8. unpow1 5.1%

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

14. Simplified 5.1%

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

15. Final simplification 5.1%

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

Alternative 11: 2.3% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 98.9%

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

9. Taylor expanded in x around 0 2.5%

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

10. Taylor expanded in x around inf 2.5%

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

    1. associate-*l/ 2.5%

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

    2. *-lft-identity 2.5%

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

    3. unpow1 2.5%

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

    4. sqr-pow 2.5%

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

    5. fabs-sqr 2.5%

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

    6. sqr-pow 2.5%

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

    7. unpow1 2.5%

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

12. Simplified 2.5%

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

13. Final simplification 2.5%

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

Reproduce

herbie shell --seed 2023227 
(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)))))))