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

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 7

Speedup: 3.3×

• 99.6% 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, 3.3× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x - 0.5 \cdot \log \pi} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (- (* x x) (* 0.5 (log PI))))
  (+
   (/ 0.5 (pow x 3.0))
   (/ (+ 1.0 (fma 0.75 (pow x -4.0) (* (pow x -6.0) 1.875))) x))))

C

double code(double x) {
	return exp(((x * x) - (0.5 * log(((double) M_PI))))) * ((0.5 / pow(x, 3.0)) + ((1.0 + fma(0.75, pow(x, -4.0), (pow(x, -6.0) * 1.875))) / x));
}

Julia

function code(x)
	return Float64(exp(Float64(Float64(x * x) - Float64(0.5 * log(pi)))) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(Float64(1.0 + fma(0.75, (x ^ -4.0), Float64((x ^ -6.0) * 1.875))) / x)))
end

Wolfram

code[x_] := N[(N[Exp[N[(N[(x * x), $MachinePrecision] - N[(0.5 * N[Log[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.75 * N[Power[x, -4.0], $MachinePrecision] + N[(N[Power[x, -6.0], $MachinePrecision] * 1.875), $MachinePrecision]), $MachinePrecision]), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]

   2. un-div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \color{blue}{\frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|x\right|}}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right) \]

   8. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{x}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right)} \]
7. Step-by-step derivation

   1. add-exp-log 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   2. log-div 100.0%

      \[\leadsto e^{\color{blue}{\log \left({\left(e^{x}\right)}^{x}\right) - \log \left(\sqrt{\pi}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   3. pow-exp 100.0%

      \[\leadsto e^{\log \color{blue}{\left(e^{x \cdot x}\right)} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   4. add-log-exp 100.0%

      \[\leadsto e^{\color{blue}{x \cdot x} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   5. pow2 100.0%

      \[\leadsto e^{\color{blue}{{x}^{2}} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{{x}^{2} - \log \left(\sqrt{\pi}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]
9. Step-by-step derivation

   1. pow2 100.0%

      \[\leadsto e^{\color{blue}{x \cdot x} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto e^{\color{blue}{x \cdot x} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]
11. Step-by-step derivation

    1. pow1/2 100.0%

       \[\leadsto e^{x \cdot x - \log \color{blue}{\left({\pi}^{0.5}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

    2. log-pow 100.0%

       \[\leadsto e^{x \cdot x - \color{blue}{0.5 \cdot \log \pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto e^{x \cdot x - \color{blue}{0.5 \cdot \log \pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (pow x 2.0)) (sqrt PI))
  (+ (/ 0.5 (pow x 3.0)) (/ (+ 1.0 (/ 0.75 (pow x 4.0))) x))))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / sqrt(((double) M_PI))) * ((0.5 / pow(x, 3.0)) + ((1.0 + (0.75 / pow(x, 4.0))) / x));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) / Math.sqrt(Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((1.0 + (0.75 / Math.pow(x, 4.0))) / x));
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / math.sqrt(math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((1.0 + (0.75 / math.pow(x, 4.0))) / x))

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(Float64(1.0 + Float64(0.75 / (x ^ 4.0))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) / sqrt(pi)) * ((0.5 / (x ^ 3.0)) + ((1.0 + (0.75 / (x ^ 4.0))) / x));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]

   2. un-div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \color{blue}{\frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|x\right|}}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right) \]

   8. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{x}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right)} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \color{blue}{\frac{0.75}{{x}^{4}}}}{x}\right) \]

8. Taylor expanded in x around inf 99.8%

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}}}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right) \]
9. Add Preprocessing

Alternative 4: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((1.0 + (0.5 / (x ^ 2.0))) / 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[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]

   2. un-div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \color{blue}{\frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|x\right|}}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right) \]

   8. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{x}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right)} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \color{blue}{\frac{0.75}{{x}^{4}}}}{x}\right) \]

8. Taylor expanded in x around inf 99.8%

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

   1. associate-*r/ 99.8%

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

   2. metadata-eval 99.8%

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

10. Simplified 99.8%

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

Alternative 5: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]

   2. un-div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \color{blue}{\frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|x\right|}}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right) \]

   8. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\color{blue}{x}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right)} \]
7. Step-by-step derivation

   1. add-exp-log 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   2. log-div 100.0%

      \[\leadsto e^{\color{blue}{\log \left({\left(e^{x}\right)}^{x}\right) - \log \left(\sqrt{\pi}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   3. pow-exp 100.0%

      \[\leadsto e^{\log \color{blue}{\left(e^{x \cdot x}\right)} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   4. add-log-exp 100.0%

      \[\leadsto e^{\color{blue}{x \cdot x} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

   5. pow2 100.0%

      \[\leadsto e^{\color{blue}{{x}^{2}} - \log \left(\sqrt{\pi}\right)} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{{x}^{2} - \log \left(\sqrt{\pi}\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, {x}^{-6} \cdot 1.875\right)}{x}\right) \]

9. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\frac{e^{{x}^{2} - \log \left(\sqrt{\pi}\right)}}{x}} \]
10. Step-by-step derivation

    1. exp-diff 99.7%

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

    2. rem-exp-log 99.7%

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

    3. associate-/l/ 99.7%

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

11. Simplified 99.7%

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

Alternative 6: 2.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 30.1%

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

   1. unpow2 30.1%

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

   2. sqr-abs 30.1%

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

   3. unpow3 30.1%

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

7. Simplified 30.1%

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

8. Taylor expanded in x around 0 1.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right) + 0.5 \cdot \left(\frac{{x}^{2}}{{\left(\left|x\right|\right)}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
9. Step-by-step derivation

   1. distribute-lft-out 1.1%

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

   2. *-commutative 1.1%

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

   3. distribute-lft-out 1.1%

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

   4. unpow3 1.1%

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

   5. sqr-abs 1.1%

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

   6. unpow2 1.1%

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

   7. associate-/r* 1.2%

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

   8. *-inverses 2.2%

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

10. Simplified 2.2%

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

    1. pow1 2.2%

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

    2. associate-*r* 2.2%

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

    3. inv-pow 2.2%

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

    4. sqrt-pow1 2.2%

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

    5. metadata-eval 2.2%

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

    6. pow-flip 2.2%

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

    7. metadata-eval 2.2%

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

12. Applied egg-rr 2.2%

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

    1. unpow1 2.2%

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

    2. +-commutative 2.2%

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

    3. rem-square-sqrt 2.2%

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

    4. fabs-sqr 2.2%

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

    5. rem-square-sqrt 2.2%

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

    6. rem-square-sqrt 2.2%

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

    7. fabs-sqr 2.2%

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

    8. rem-square-sqrt 2.2%

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

14. Simplified 2.2%

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

Alternative 7: 2.3% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (/ 0.5 x)))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (0.5 / x);
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (0.5 / x);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (0.5 / x)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(0.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (0.5 / x);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.5 / 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%

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

5. Taylor expanded in x around 0 30.1%

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

   1. unpow2 30.1%

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

   2. sqr-abs 30.1%

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

   3. unpow3 30.1%

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

7. Simplified 30.1%

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

8. Taylor expanded in x around 0 1.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right) + 0.5 \cdot \left(\frac{{x}^{2}}{{\left(\left|x\right|\right)}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
9. Step-by-step derivation

   1. distribute-lft-out 1.1%

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

   2. *-commutative 1.1%

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

   3. distribute-lft-out 1.1%

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

   4. unpow3 1.1%

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

   5. sqr-abs 1.1%

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

   6. unpow2 1.1%

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

   7. associate-/r* 1.2%

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

   8. *-inverses 2.2%

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

10. Simplified 2.2%

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

    1. pow1 2.2%

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

    2. associate-*r* 2.2%

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

    3. inv-pow 2.2%

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

    4. sqrt-pow1 2.2%

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

    5. metadata-eval 2.2%

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

    6. pow-flip 2.2%

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

    7. metadata-eval 2.2%

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

12. Applied egg-rr 2.2%

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

    1. unpow1 2.2%

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

    2. +-commutative 2.2%

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

    3. rem-square-sqrt 2.2%

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

    4. fabs-sqr 2.2%

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

    5. rem-square-sqrt 2.2%

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

    6. rem-square-sqrt 2.2%

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

    7. fabs-sqr 2.2%

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

    8. rem-square-sqrt 2.2%

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

14. Simplified 2.2%

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

15. Taylor expanded in x around inf 2.2%

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

    1. associate-*r* 2.2%

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

    2. associate-*r/ 2.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} \cdot \sqrt{\frac{1}{\pi}} \]

    3. metadata-eval 2.2%

       \[\leadsto \frac{\color{blue}{0.5}}{x} \cdot \sqrt{\frac{1}{\pi}} \]

    4. *-commutative 2.2%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x}} \]

17. Simplified 2.2%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x}} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))))