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

Percentage Accurate: 100.0% → 99.9%

Time: 17.5s

Alternatives: 9

Speedup: 4.0×

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

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp (* (pow x 2.0) 3.0)) 0.3333333333333333) (cbrt (pow PI 1.5)))
  (fma
   0.75
   (log (exp (pow x -5.0)))
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	return (pow(exp((pow(x, 2.0) * 3.0)), 0.3333333333333333) / cbrt(pow(((double) M_PI), 1.5))) * fma(0.75, log(exp(pow(x, -5.0))), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64((exp(Float64((x ^ 2.0) * 3.0)) ^ 0.3333333333333333) / cbrt((pi ^ 1.5))) * fma(0.75, log(exp((x ^ -5.0))), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[N[(N[Power[x, 2.0], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Log[N[Exp[N[Power[x, -5.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[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%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
9. Step-by-step derivation

   1. add-cbrt-cube 100.0%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(e^{x \cdot x} \cdot e^{x \cdot x}\right) \cdot e^{x \cdot x}}}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. pow1/3 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\left(e^{x \cdot x} \cdot e^{x \cdot x}\right) \cdot e^{x \cdot x}\right)}^{0.3333333333333333}}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow3 100.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(e^{x \cdot x}\right)}^{3}\right)}}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. pow-to-exp 100.0%

      \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(e^{x \cdot x}\right) \cdot 3}\right)}}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{\color{blue}{\left(x \cdot x\right)} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. pow2 100.0%

      \[\leadsto \frac{{\left(e^{\color{blue}{{x}^{2}} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. inv-pow 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow-pow 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{x}}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{\color{blue}{-5}}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\log \left(e^{{x}^{-5}}\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
13. Step-by-step derivation

    1. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. *-un-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. log-prod 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. inv-pow 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. pow-pow 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    8. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    9. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    10. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    11. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
15. Step-by-step derivation

    1. +-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

16. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
17. Add Preprocessing

Alternative 2: 99.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right) + 0.75 \cdot {x}^{-5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp (* (pow x 2.0) 3.0)) 0.3333333333333333) (sqrt PI))
  (+
   (fma 1.875 (pow x -7.0) (/ (fma 0.5 (pow x -2.0) 1.0) x))
   (* 0.75 (pow x -5.0)))))

C

double code(double x) {
	return (pow(exp((pow(x, 2.0) * 3.0)), 0.3333333333333333) / sqrt(((double) M_PI))) * (fma(1.875, pow(x, -7.0), (fma(0.5, pow(x, -2.0), 1.0) / x)) + (0.75 * pow(x, -5.0)));
}

Julia

function code(x)
	return Float64(Float64((exp(Float64((x ^ 2.0) * 3.0)) ^ 0.3333333333333333) / sqrt(pi)) * Float64(fma(1.875, (x ^ -7.0), Float64(fma(0.5, (x ^ -2.0), 1.0) / x)) + Float64(0.75 * (x ^ -5.0))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[N[(N[Power[x, 2.0], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. div-inv 100.0%

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

    5. fma-define 100.0%

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

    6. pow2 100.0%

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

    7. pow-flip 100.0%

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

    8. metadata-eval 100.0%

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

    9. add-sqr-sqrt 100.0%

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

    10. fabs-sqr 100.0%

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

    11. add-sqr-sqrt 100.0%

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

14. Applied egg-rr 100.0%

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

    1. add-cbrt-cube 100.0%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(e^{x \cdot x} \cdot e^{x \cdot x}\right) \cdot e^{x \cdot x}}}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. pow1/3 100.0%

       \[\leadsto \frac{\color{blue}{{\left(\left(e^{x \cdot x} \cdot e^{x \cdot x}\right) \cdot e^{x \cdot x}\right)}^{0.3333333333333333}}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow3 100.0%

       \[\leadsto \frac{{\color{blue}{\left({\left(e^{x \cdot x}\right)}^{3}\right)}}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. pow-to-exp 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(e^{x \cdot x}\right) \cdot 3}\right)}}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{\left(x \cdot x\right)} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. pow2 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{{x}^{2}} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

16. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right) + 0.75 \cdot {x}^{-5}\right) \]
17. Add Preprocessing

Alternative 3: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{\color{blue}{x}}^{\left(-1 \cdot 5\right)}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{\color{blue}{-5}}}\right), \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\left(1 + \frac{0.5}{{x}^{2}}\right) + \frac{0.75 + \frac{1.875}{{x}^{2}}}{{x}^{4}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (/
   (+
    (+ 1.0 (/ 0.5 (pow x 2.0)))
    (/ (+ 0.75 (/ 1.875 (pow x 2.0))) (pow x 4.0)))
   x)))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * (((1.0 + (0.5 / pow(x, 2.0))) + ((0.75 + (1.875 / pow(x, 2.0))) / pow(x, 4.0))) / x);
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * (((1.0 + (0.5 / Math.pow(x, 2.0))) + ((0.75 + (1.875 / Math.pow(x, 2.0))) / Math.pow(x, 4.0))) / x);
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (((1.0 + (0.5 / math.pow(x, 2.0))) + ((0.75 + (1.875 / math.pow(x, 2.0))) / math.pow(x, 4.0))) / x)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) + Float64(Float64(0.75 + Float64(1.875 / (x ^ 2.0))) / (x ^ 4.0))) / x))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * (((1.0 + (0.5 / (x ^ 2.0))) + ((0.75 + (1.875 / (x ^ 2.0))) / (x ^ 4.0))) / x);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 + N[(1.875 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 4.0], $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%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. div-inv 100.0%

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

    5. fma-define 100.0%

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

    6. pow2 100.0%

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

    7. pow-flip 100.0%

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

    8. metadata-eval 100.0%

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

    9. add-sqr-sqrt 100.0%

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

    10. fabs-sqr 100.0%

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

    11. add-sqr-sqrt 100.0%

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

14. Applied egg-rr 100.0%

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

15. Taylor expanded in x around -inf 100.0%

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

    1. associate-*r/ 100.0%

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

17. Simplified 100.0%

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

Alternative 5: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. div-inv 100.0%

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

    5. fma-define 100.0%

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

    6. pow2 100.0%

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

    7. pow-flip 100.0%

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

    8. metadata-eval 100.0%

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

    9. add-sqr-sqrt 100.0%

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

    10. fabs-sqr 100.0%

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

    11. add-sqr-sqrt 100.0%

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

14. Applied egg-rr 100.0%

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

15. Taylor expanded in x around inf 99.5%

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

    1. associate-+r+ 99.5%

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

    2. +-commutative 99.5%

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

    3. associate-*r/ 99.5%

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

    4. metadata-eval 99.5%

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

17. Simplified 99.5%

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

Alternative 6: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

15. Simplified 99.4%

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

    1. expm1-log1p-u 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{{x}^{2} \cdot \left|x\right|}\right)\right)}\right) \]

    2. expm1-undefine 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{2} \cdot \left|x\right|}\right)} - 1\right)}\right) \]

    3. pow2 99.4%

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

    4. associate-/r* 99.4%

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

    5. add-sqr-sqrt 99.4%

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

    6. fabs-sqr 99.4%

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

    7. add-sqr-sqrt 99.4%

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

    8. associate-/r* 99.4%

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

    9. pow3 99.4%

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

17. Applied egg-rr 99.4%

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

    1. log1p-undefine 99.4%

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

    2. rem-exp-log 99.4%

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

    3. +-commutative 99.4%

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

    4. associate--l+ 99.4%

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

    5. metadata-eval 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{{x}^{3}} + \color{blue}{0}\right)\right) \]

    6. +-commutative 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\left(0 + \frac{0.5}{{x}^{3}}\right)}\right) \]

    7. +-lft-identity 99.4%

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

19. Simplified 99.4%

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

Alternative 7: 99.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	return (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(Float64(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[Exp[N[(x * x), $MachinePrecision]], $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%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. div-inv 100.0%

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

    5. fma-define 100.0%

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

    6. pow2 100.0%

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

    7. pow-flip 100.0%

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

    8. metadata-eval 100.0%

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

    9. add-sqr-sqrt 100.0%

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

    10. fabs-sqr 100.0%

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

    11. add-sqr-sqrt 100.0%

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

14. Applied egg-rr 100.0%

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

15. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

17. Simplified 99.4%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}} \]
18. Add Preprocessing

Alternative 8: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 / 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. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. div-inv 100.0%

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

    5. fma-define 100.0%

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

    6. pow2 100.0%

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

    7. pow-flip 100.0%

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

    8. metadata-eval 100.0%

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

    9. add-sqr-sqrt 100.0%

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

    10. fabs-sqr 100.0%

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

    11. add-sqr-sqrt 100.0%

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

14. Applied egg-rr 100.0%

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

15. Taylor expanded in x around inf 99.4%

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

Alternative 9: 2.3% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{\log 1 + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{{x}^{2} \cdot 3}\right)}^{0.3333333333333333}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \log \left(e^{{x}^{-5}}\right), \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right), \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 0 + {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{0 + {x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
11. Step-by-step derivation

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

14. Taylor expanded in x around 0 2.4%

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

15. Final simplification 2.4%

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

Reproduce

herbie shell --seed 2024108 
(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)))))))