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

Percentage Accurate: 100.0% → 100.0%

Time: 12.6s

Alternatives: 7

Speedup: 3.3×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision], 5.0], $MachinePrecision] + N[(1.875 * N[Log[N[Exp[N[Power[x, -7.0], $MachinePrecision]], $MachinePrecision]], $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-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(0.75, pow((1.0 / fabs(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 * x)) / sqrt(pi)) * fma(0.75, (Float64(1.0 / abs(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[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision], 5.0], $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. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

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. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((0.75 / Math.pow(x, 5.0)) + ((1.0 / Math.abs(x)) + (1.875 * (1.0 / Math.pow(x, 7.0))))));
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((0.75 / math.pow(x, 5.0)) + ((1.0 / math.fabs(x)) + (1.875 * (1.0 / math.pow(x, 7.0))))))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(1.0 / abs(x)) + Float64(1.875 * Float64(1.0 / (x ^ 7.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * ((0.5 / (x ^ 3.0)) + ((0.75 / (x ^ 5.0)) + ((1.0 / abs(x)) + (1.875 * (1.0 / (x ^ 7.0))))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

    1. div-inv 100.0%

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

    2. pow2 100.0%

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

    3. *-commutative 100.0%

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

    4. pow2 100.0%

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

13. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. associate-/l/ 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. fabs-sqr 100.0%

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

    6. rem-square-sqrt 100.0%

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

    7. associate-/r* 100.0%

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

    8. unpow2 100.0%

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

    9. unpow3 100.0%

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

15. Simplified 100.0%

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

Alternative 4: 99.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

12. Taylor expanded in x around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. rem-square-sqrt 99.4%

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

    3. fabs-sqr 99.4%

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

    4. rem-square-sqrt 99.4%

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

    5. associate-/r* 99.4%

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

    6. unpow2 99.4%

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

    7. unpow3 99.4%

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

    8. metadata-eval 99.4%

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

    9. associate-*r/ 99.4%

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

    10. fma-define 99.4%

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

    11. exp-to-pow 99.4%

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

    12. *-commutative 99.4%

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

    13. exp-neg 99.4%

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

    14. *-commutative 99.4%

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

    15. distribute-rgt-neg-in 99.4%

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

    16. metadata-eval 99.4%

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

    17. exp-to-pow 99.4%

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

    18. rem-square-sqrt 99.4%

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

    19. fabs-sqr 99.4%

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

    20. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

Alternative 5: 99.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

12. Taylor expanded in x around inf 99.4%

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

    1. rem-square-sqrt 99.4%

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

    2. fabs-sqr 99.4%

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

    3. rem-square-sqrt 99.4%

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

    4. associate-*r/ 99.4%

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

    5. metadata-eval 99.4%

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

    6. associate-/r* 99.4%

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

    7. rem-square-sqrt 99.4%

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

    8. fabs-sqr 99.4%

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

    9. rem-square-sqrt 99.4%

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

    10. associate-/r* 99.4%

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

    11. unpow2 99.4%

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

    12. unpow3 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

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

Alternative 6: 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%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

12. Taylor expanded in x around inf 99.4%

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

    1. rem-square-sqrt 99.4%

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

    2. fabs-sqr 99.4%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

Alternative 7: 1.8% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 1.9%

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

   1. associate-*l/ 1.9%

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

   2. *-un-lft-identity 1.9%

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

   3. pow2 1.9%

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

   4. associate-/r* 1.9%

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

   5. inv-pow 1.9%

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

   6. sqrt-pow1 1.9%

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

   7. metadata-eval 1.9%

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

   8. pow2 1.9%

      \[\leadsto 0.5 \cdot \frac{\frac{{\pi}^{-0.5}}{\color{blue}{{x}^{2}}}}{\left|x\right|} \]

7. Applied egg-rr 1.9%

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

   1. add-cube-cbrt 1.9%

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

   2. pow3 1.9%

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

   3. pow2 1.9%

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

   4. associate-/l/ 1.9%

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

   5. cbrt-div 1.9%

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

   6. sqr-abs 1.9%

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

   7. cube-mult 1.9%

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

   8. pow3 1.9%

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

   9. add-cbrt-cube 1.9%

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

   10. add-sqr-sqrt 1.9%

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

   11. fabs-sqr 1.9%

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

   12. add-sqr-sqrt 1.9%

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

9. Applied egg-rr 1.9%

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

    1. div-inv 1.9%

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

    2. unpow-prod-down 1.9%

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

    3. rem-cube-cbrt 1.9%

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

    4. inv-pow 1.9%

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

    5. pow-pow 1.9%

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

    6. metadata-eval 1.9%

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

11. Applied egg-rr 1.9%

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

Reproduce

herbie shell --seed 2024123 
(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)))))))