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

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 10

Speedup: 4.2×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   3. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{4}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   4. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   6. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 4\right)}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{e^{\mathsf{log1p}\left({x}^{\color{blue}{-4}}\right)} - 1}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({x}^{-4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-4}\right)\right)}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. fabs-sqr 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. pow1 100.0%

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

   10. pow-div 100.0%

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

   11. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (/ (exp (* x x)) (fabs x)) (cbrt (pow PI 1.5)))
  (+ 1.0 (+ (/ 1.875 (pow x 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))))

C

double code(double x) {
	return ((exp((x * x)) / fabs(x)) / cbrt(pow(((double) M_PI), 1.5))) * (1.0 + ((1.875 / pow(x, 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Java

public static double code(double x) {
	return ((Math.exp((x * x)) / Math.abs(x)) / Math.cbrt(Math.pow(Math.PI, 1.5))) * (1.0 + ((1.875 / Math.pow(x, 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Julia

function code(x)
	return Float64(Float64(Float64(exp(Float64(x * x)) / abs(x)) / cbrt((pi ^ 1.5))) * Float64(1.0 + Float64(Float64(1.875 / (x ^ 6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))))
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

8. Final simplification 100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

Alternative 3: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 99.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))))
  (log1p (expm1 (/ x (sqrt PI))))))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) * log1p(expm1((x / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))) * Math.log1p(Math.expm1((x / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return (1.0 + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))) * math.log1p(math.expm1((x / math.sqrt(math.pi))))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) * log1p(expm1(Float64(x / sqrt(pi)))))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 59.1%

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

   1. unpow2 59.1%

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

9. Simplified 59.1%

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

10. Taylor expanded in x around inf 59.1%

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

    1. unpow2 59.1%

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

    2. associate-/l* 5.8%

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

    3. unpow1 5.8%

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

    4. sqr-pow 5.8%

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

    5. metadata-eval 5.8%

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

    6. unpow1/2 5.8%

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

    7. metadata-eval 5.8%

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

    8. unpow1/2 5.8%

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

    9. fabs-sqr 5.8%

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

    10. unpow1/2 5.8%

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

    11. metadata-eval 5.8%

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

    12. unpow1/2 5.8%

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

    13. metadata-eval 5.8%

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

    14. sqr-pow 5.8%

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

    15. unpow1 5.8%

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

12. Simplified 5.8%

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

    1. log1p-expm1-u 99.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{x}{\frac{x}{x}}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. associate-/r/ 99.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\frac{x}{x} \cdot x}}{\sqrt{\pi}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. *-inverses 99.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1} \cdot x}{\sqrt{\pi}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. *-un-lft-identity 99.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

15. Final simplification 99.7%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi}}\right)\right) \]

Alternative 6: 52.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 59.1%

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

   1. unpow2 59.1%

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

9. Simplified 59.1%

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

    1. add-sqr-sqrt 59.1%

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

    2. fabs-sqr 59.1%

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

    3. add-sqr-sqrt 59.1%

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

    4. div-inv 59.1%

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

    5. +-commutative 59.1%

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

    6. fma-def 59.1%

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

11. Applied egg-rr 59.1%

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

    1. un-div-inv 59.1%

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

    2. fma-udef 59.1%

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

    3. metadata-eval 59.1%

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

    4. *-un-lft-identity 59.1%

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

    5. frac-add 5.8%

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

    6. *-inverses 5.8%

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

    7. +-commutative 5.8%

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

    8. *-inverses 5.8%

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

    9. frac-add 59.1%

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

    10. metadata-eval 59.1%

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

    11. *-inverses 59.1%

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

    12. clear-num 59.1%

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

    13. div-inv 59.1%

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

    14. associate-/r/ 59.1%

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

    15. *-inverses 59.1%

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

    16. *-un-lft-identity 59.1%

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

13. Applied egg-rr 59.1%

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

14. Final simplification 59.1%

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

Alternative 7: 52.4% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 59.1%

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

   1. unpow2 59.1%

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

9. Simplified 59.1%

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

10. Taylor expanded in x around inf 59.1%

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

    1. unpow2 59.1%

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

    2. associate-/l* 5.8%

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

    3. unpow1 5.8%

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

    4. sqr-pow 5.8%

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

    5. metadata-eval 5.8%

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

    6. unpow1/2 5.8%

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

    7. metadata-eval 5.8%

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

    8. unpow1/2 5.8%

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

    9. fabs-sqr 5.8%

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

    10. unpow1/2 5.8%

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

    11. metadata-eval 5.8%

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

    12. unpow1/2 5.8%

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

    13. metadata-eval 5.8%

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

    14. sqr-pow 5.8%

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

    15. unpow1 5.8%

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

12. Simplified 5.8%

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

    1. add-sqr-sqrt 5.8%

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

    2. sqrt-unprod 59.1%

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

    3. associate-/r/ 59.1%

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

    4. *-inverses 59.1%

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

    5. *-un-lft-identity 59.1%

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

    6. associate-/r/ 59.1%

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

    7. *-inverses 59.1%

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

    8. *-un-lft-identity 59.1%

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

    9. frac-times 59.1%

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

    10. add-sqr-sqrt 59.1%

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

14. Applied egg-rr 59.1%

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

15. Final simplification 59.1%

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

Alternative 8: 5.4% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 59.1%

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

   1. unpow2 59.1%

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

9. Simplified 59.1%

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

    1. add-sqr-sqrt 59.1%

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

    2. fabs-sqr 59.1%

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

    3. add-sqr-sqrt 59.1%

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

    4. div-inv 59.1%

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

    5. +-commutative 59.1%

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

    6. fma-def 59.1%

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

11. Applied egg-rr 59.1%

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

12. Taylor expanded in x around inf 5.8%

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

13. Final simplification 5.8%

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

Alternative 9: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 2.2%

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

   1. expm1-log1p-u 2.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   2. expm1-udef 1.7%

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

   3. div-inv 1.7%

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

   4. clear-num 1.7%

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

   5. associate-*l/ 1.7%

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

   6. *-un-lft-identity 1.7%

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

   7. pow1/2 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   8. pow-flip 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   9. metadata-eval 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   10. add-sqr-sqrt 1.7%

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

   11. fabs-sqr 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   12. add-sqr-sqrt 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   13. /-rgt-identity 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\color{blue}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Applied egg-rr 1.7%

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

    1. expm1-def 2.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 2.2%

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

11. Simplified 2.2%

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

12. Taylor expanded in x around inf 2.2%

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

    1. *-commutative 2.2%

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

    2. associate-*r* 2.2%

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

    3. distribute-rgt-out 2.2%

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

    4. unpow-1 2.2%

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

    5. metadata-eval 2.2%

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

    6. pow-sqr 2.2%

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

    7. rem-sqrt-square 2.2%

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

    8. metadata-eval 2.2%

       \[\leadsto \left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) \]

    9. pow-sqr 2.2%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \cdot \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) \]

    10. fabs-sqr 2.2%

        \[\leadsto \color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)} \cdot \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) \]

    11. pow-sqr 2.2%

        \[\leadsto \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \cdot \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) \]

    12. metadata-eval 2.2%

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

    13. associate-*r/ 2.2%

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

    14. metadata-eval 2.2%

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

14. Simplified 2.2%

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

15. Final simplification 2.2%

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

Alternative 10: 1.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.875 (/ (pow PI -0.5) (pow x 7.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 2.2%

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

   1. expm1-log1p-u 2.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   2. expm1-udef 1.7%

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

   3. div-inv 1.7%

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

   4. clear-num 1.7%

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

   5. associate-*l/ 1.7%

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

   6. *-un-lft-identity 1.7%

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

   7. pow1/2 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   8. pow-flip 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   9. metadata-eval 1.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   10. add-sqr-sqrt 1.7%

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

   11. fabs-sqr 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   12. add-sqr-sqrt 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

   13. /-rgt-identity 1.7%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{\color{blue}{x}}\right)} - 1\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Applied egg-rr 1.7%

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

    1. expm1-def 2.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. expm1-log1p 2.2%

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

11. Simplified 2.2%

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

12. Taylor expanded in x around 0 1.7%

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

    1. associate-*l/ 1.7%

       \[\leadsto 1.875 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{7}}} \]

    2. *-lft-identity 1.7%

       \[\leadsto 1.875 \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{{x}^{7}} \]

    3. unpow-1 1.7%

       \[\leadsto 1.875 \cdot \frac{\sqrt{\color{blue}{{\pi}^{-1}}}}{{x}^{7}} \]

    4. metadata-eval 1.7%

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

    5. pow-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}}{{x}^{7}} \]

    6. rem-sqrt-square 1.7%

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

    7. metadata-eval 1.7%

       \[\leadsto 1.875 \cdot \frac{\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|}{{x}^{7}} \]

    8. pow-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right|}{{x}^{7}} \]

    9. fabs-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}}{{x}^{7}} \]

    10. pow-sqr 1.7%

        \[\leadsto 1.875 \cdot \frac{\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}}{{x}^{7}} \]

    11. metadata-eval 1.7%

        \[\leadsto 1.875 \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{{x}^{7}} \]

14. Simplified 1.7%

    \[\leadsto \color{blue}{1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}}} \]

15. Final simplification 1.7%

    \[\leadsto 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}} \]

Reproduce

herbie shell --seed 2023240 
(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)))))))