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

Percentage Accurate: 100.0% → 99.9%

Time: 19.4s

Alternatives: 10

Speedup: 3.5×

• 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: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-div 100.0%

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

   3. sqrt-pow1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

   6. sqr-abs 100.0%

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

   7. sqrt-div 100.0%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow2 100.0%

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

   11. sqr-abs 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. rem-square-sqrt 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-*r* 100.0%

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

   6. unpow3 100.0%

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

   7. pow-plus 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 1.6%

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

   2. *-un-lft-identity 1.6%

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

   3. log-prod 1.6%

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

   4. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   9. *-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0 + \sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (* (pow x -5.0) (+ 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)) + (pow(x, -5.0) * (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)) + (Math.pow(x, -5.0) * (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)) + (math.pow(x, -5.0) * (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((x ^ -5.0) * Float64(0.75 + Float64(Float64(1.875 / x) / x)))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + ((x ^ -5.0) * (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[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(N[(1.875 / x), $MachinePrecision] / x), $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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-div 100.0%

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

   3. sqrt-pow1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

   6. sqr-abs 100.0%

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

   7. sqrt-div 100.0%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow2 100.0%

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

   11. sqr-abs 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. rem-square-sqrt 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-*r* 100.0%

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

   6. unpow3 100.0%

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

   7. pow-plus 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 1.6%

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

   2. *-un-lft-identity 1.6%

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

   3. log-prod 1.6%

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

   4. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   9. *-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0 + \sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 99.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-div 100.0%

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

   3. sqrt-pow1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

   6. sqr-abs 100.0%

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

   7. sqrt-div 100.0%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow2 100.0%

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

   11. sqr-abs 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. rem-square-sqrt 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-*r* 100.0%

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

   6. unpow3 100.0%

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

   7. pow-plus 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 1.6%

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

   2. *-un-lft-identity 1.6%

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

   3. log-prod 1.6%

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

   4. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   9. *-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0 + \sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around inf 100.0%

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

    1. unpow2 99.2%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

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

Alternative 5: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * (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(0.75 / (x ^ 5.0))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * (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[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

9. Taylor expanded in x around inf 99.2%

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

    1. unpow2 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 6: 51.2% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-div 100.0%

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

   3. sqrt-pow1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

   6. sqr-abs 100.0%

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

   7. sqrt-div 100.0%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow2 100.0%

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

   11. sqr-abs 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. rem-square-sqrt 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-*r* 100.0%

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

   6. unpow3 100.0%

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

   7. pow-plus 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 1.6%

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

   2. *-un-lft-identity 1.6%

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

   3. log-prod 1.6%

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

   4. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   7. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

   9. *-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{\left(\left|x\right|\right)}^{6}}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0 + \sqrt{\pi} \cdot x}} \cdot \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around 0 50.1%

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

    1. unpow2 50.1%

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

14. Simplified 50.1%

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

15. Final simplification 50.1%

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

Alternative 7: 51.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * ((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(0.75 / (x ^ 5.0))) * Float64(Float64(1.0 + Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * ((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[(0.75 / N[Power[x, 5.0], $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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

9. Taylor expanded in x around 0 50.1%

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

    1. unpow2 50.1%

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

11. Simplified 50.1%

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

12. Final simplification 50.1%

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

Alternative 8: 51.3% accurate, 10.3× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

9. Taylor expanded in x around 0 50.1%

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

    1. unpow2 50.1%

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

11. Simplified 50.1%

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

12. Taylor expanded in x around inf 50.1%

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

    1. associate-*r* 50.1%

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

    2. distribute-rgt-out 50.1%

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

    3. associate-*r/ 50.1%

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

    4. metadata-eval 50.1%

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

    5. unpow1 50.1%

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

    6. sqr-pow 50.1%

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

    7. fabs-sqr 50.1%

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

    8. sqr-pow 50.1%

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

    9. unpow1 50.1%

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

    10. unpow2 50.1%

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

    11. unpow1 50.1%

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

    12. sqr-pow 50.1%

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

    13. fabs-sqr 50.1%

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

    14. sqr-pow 50.1%

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

    15. unpow1 50.1%

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

14. Simplified 50.1%

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

15. Final simplification 50.1%

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

Alternative 9: 51.3% accurate, 10.5× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

9. Taylor expanded in x around 0 50.1%

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

    1. unpow2 50.1%

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

11. Simplified 50.1%

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

12. Taylor expanded in x around inf 50.1%

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

    1. *-commutative 50.1%

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

    2. unpow2 50.1%

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

    3. unpow1 50.1%

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

    4. sqr-pow 50.1%

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

    5. fabs-sqr 50.1%

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

    6. sqr-pow 50.1%

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

    7. unpow1 50.1%

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

14. Simplified 50.1%

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

15. Final simplification 50.1%

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

Alternative 10: 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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

9. Taylor expanded in x around 0 2.4%

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

10. Taylor expanded in x around inf 2.4%

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

    1. associate-*l/ 2.4%

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

    2. *-lft-identity 2.4%

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

    3. unpow1 2.4%

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

    4. sqr-pow 2.4%

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

    5. fabs-sqr 2.4%

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

    6. sqr-pow 2.4%

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

    7. unpow1 2.4%

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

12. Simplified 2.4%

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

13. Final simplification 2.4%

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

Reproduce

herbie shell --seed 2023279 
(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)))))))