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

Percentage Accurate: 100.0% → 100.0%

Time: 25.4s

Alternatives: 8

Speedup: 3.4×

• 99.3% 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, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. pow1/2 100.0%

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

    2. metadata-eval 100.0%

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

    3. metadata-eval 100.0%

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

    4. pow-pow 100.0%

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

    5. metadata-eval 100.0%

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

14. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

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

16. Simplified 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. fabs-sqr 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. add-sqr-sqrt 100.0%

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

    5. associate-/l* 100.0%

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

    6. add-sqr-sqrt 100.0%

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

    7. hypot-1-def 100.0%

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

    8. sqrt-div 100.0%

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

    9. sqrt-prod 100.0%

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

    10. add-sqr-sqrt 100.0%

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

    11. add-sqr-sqrt 100.0%

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

    12. hypot-1-def 100.0%

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

    13. sqrt-div 100.0%

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

    14. sqrt-prod 100.0%

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

    15. add-sqr-sqrt 100.0%

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

18. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. hypot-undefine 100.0%

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

    3. hypot-undefine 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. metadata-eval 100.0%

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

    6. associate-*r/ 100.0%

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

    7. associate-*l/ 100.0%

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

    8. rem-square-sqrt 100.0%

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

    9. associate-/r* 100.0%

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

    10. unpow2 100.0%

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

20. Simplified 100.0%

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

    1. pow2 100.0%

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

    2. associate-/r* 100.0%

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

    3. div-inv 100.0%

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

22. Applied egg-rr 100.0%

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

    1. un-div-inv 100.0%

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

24. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. pow1/2 100.0%

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

    2. metadata-eval 100.0%

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

    3. metadata-eval 100.0%

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

    4. pow-pow 100.0%

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

    5. metadata-eval 100.0%

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

14. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

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

16. Simplified 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. fabs-sqr 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. add-sqr-sqrt 100.0%

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

    5. associate-/l* 100.0%

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

    6. add-sqr-sqrt 100.0%

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

    7. hypot-1-def 100.0%

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

    8. sqrt-div 100.0%

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

    9. sqrt-prod 100.0%

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

    10. add-sqr-sqrt 100.0%

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

    11. add-sqr-sqrt 100.0%

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

    12. hypot-1-def 100.0%

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

    13. sqrt-div 100.0%

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

    14. sqrt-prod 100.0%

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

    15. add-sqr-sqrt 100.0%

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

18. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. hypot-undefine 100.0%

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

    3. hypot-undefine 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. metadata-eval 100.0%

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

    6. associate-*r/ 100.0%

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

    7. associate-*l/ 100.0%

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

    8. rem-square-sqrt 100.0%

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

    9. associate-/r* 100.0%

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

    10. unpow2 100.0%

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

20. Simplified 100.0%

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

    1. pow2 100.0%

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

    2. associate-/r* 100.0%

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

    3. div-inv 100.0%

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

22. Applied egg-rr 100.0%

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

    1. un-div-inv 100.0%

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

24. Applied egg-rr 100.0%

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

    1. pow1/3 100.0%

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

    2. pow-pow 100.0%

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

    3. metadata-eval 100.0%

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

    4. pow1/2 100.0%

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

    5. /-rgt-identity 100.0%

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

    6. clear-num 100.0%

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

    7. pow1/2 100.0%

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

    8. pow-flip 100.0%

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

    9. metadata-eval 100.0%

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

26. Applied egg-rr 100.0%

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

27. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. pow1/2 100.0%

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

    2. metadata-eval 100.0%

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

    3. metadata-eval 100.0%

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

    4. pow-pow 100.0%

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

    5. metadata-eval 100.0%

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

14. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

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

16. Simplified 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. fabs-sqr 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. add-sqr-sqrt 100.0%

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

    5. associate-/l* 100.0%

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

    6. add-sqr-sqrt 100.0%

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

    7. hypot-1-def 100.0%

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

    8. sqrt-div 100.0%

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

    9. sqrt-prod 100.0%

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

    10. add-sqr-sqrt 100.0%

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

    11. add-sqr-sqrt 100.0%

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

    12. hypot-1-def 100.0%

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

    13. sqrt-div 100.0%

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

    14. sqrt-prod 100.0%

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

    15. add-sqr-sqrt 100.0%

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

18. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. hypot-undefine 100.0%

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

    3. hypot-undefine 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. metadata-eval 100.0%

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

    6. associate-*r/ 100.0%

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

    7. associate-*l/ 100.0%

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

    8. rem-square-sqrt 100.0%

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

    9. associate-/r* 100.0%

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

    10. unpow2 100.0%

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

20. Simplified 100.0%

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

    1. pow2 100.0%

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

    2. associate-/r* 100.0%

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

    3. div-inv 100.0%

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

22. Applied egg-rr 100.0%

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

    1. un-div-inv 100.0%

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

24. Applied egg-rr 100.0%

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

    1. pow1/3 100.0%

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

    2. pow-pow 100.0%

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

    3. metadata-eval 100.0%

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

    4. pow1/2 100.0%

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

    5. expm1-log1p-u 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]

    6. expm1-undefine 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi}\right)} - 1}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]

26. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi}\right)} - 1}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]
27. Step-by-step derivation

    1. sub-neg 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi}\right)} + \left(-1\right)}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]

    2. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{e^{\mathsf{log1p}\left(\sqrt{\pi}\right)} + \color{blue}{-1}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]

    3. +-commutative 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{-1 + e^{\mathsf{log1p}\left(\sqrt{\pi}\right)}}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\frac{0.5}{x}}{x}}{x}\right)\right) \]

    4. log1p-undefine 100.0%

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

    5. rem-exp-log 100.0%

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

    6. associate-+r+ 100.0%

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

    7. metadata-eval 100.0%

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

    8. metadata-eval 100.0%

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

    9. associate--r- 100.0%

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

    10. neg-sub0 100.0%

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

    11. neg-sub0 100.0%

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

    12. remove-double-neg 100.0%

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

28. Simplified 100.0%

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

29. Final simplification 100.0%

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

Alternative 4: 99.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.7%

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

    1. fma-undefine 99.7%

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

    2. +-commutative 99.7%

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

    3. add-sqr-sqrt 99.7%

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

    4. fabs-sqr 99.7%

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

    5. add-sqr-sqrt 99.7%

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

15. Applied egg-rr 99.7%

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

Alternative 5: 99.4% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * 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[(0.75 * N[Power[x, -5.0], $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%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.7%

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

    1. fma-undefine 99.7%

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

    2. +-commutative 99.7%

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

    3. add-sqr-sqrt 99.7%

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

    4. fabs-sqr 99.7%

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

    5. add-sqr-sqrt 99.7%

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

15. Applied egg-rr 99.7%

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

16. Taylor expanded in x around inf 99.7%

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

Alternative 6: 98.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{x}{\sqrt{\pi}}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (exp (/ x (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 2.3%

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

14. Taylor expanded in x around inf 2.3%

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

    1. associate-*r/ 2.3%

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

    2. *-rgt-identity 2.3%

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

16. Simplified 2.3%

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

    1. frac-2neg 2.3%

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

    2. div-inv 2.3%

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

    3. sqrt-div 2.3%

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

    4. metadata-eval 2.3%

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

    5. distribute-neg-frac 2.3%

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

    6. metadata-eval 2.3%

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

    7. add-sqr-sqrt 2.3%

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

    8. fabs-sqr 2.3%

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

    9. add-sqr-sqrt 2.3%

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

18. Applied egg-rr 2.3%

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

    1. distribute-frac-neg2 2.3%

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

    2. associate-*l/ 2.3%

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

    3. mul-1-neg 2.3%

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

    4. remove-double-neg 2.3%

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

20. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt{\pi}}} \]
21. Step-by-step derivation

    1. add-sqr-sqrt 2.3%

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

    2. associate-*r/ 2.3%

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

    3. sqrt-div 2.3%

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

    4. *-commutative 2.3%

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

    5. sqrt-div 2.3%

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

    6. metadata-eval 2.3%

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

    7. div-inv 2.3%

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

    8. add-log-exp 1.7%

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

    9. *-un-lft-identity 1.7%

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

    10. associate-*l/ 1.7%

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

    11. metadata-eval 1.7%

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

    12. sqrt-div 1.7%

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

    13. sqrt-div 1.7%

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

    14. associate-*r/ 1.7%

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

    15. add-sqr-sqrt 1.7%

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

22. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{\log \left(e^{\frac{x}{\sqrt{\pi}}}\right)} \]
23. Add Preprocessing

Alternative 7: 51.9% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 2.3%

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

14. Taylor expanded in x around inf 2.3%

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

    1. associate-*r/ 2.3%

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

    2. *-rgt-identity 2.3%

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

16. Simplified 2.3%

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

    1. frac-2neg 2.3%

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

    2. div-inv 2.3%

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

    3. sqrt-div 2.3%

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

    4. metadata-eval 2.3%

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

    5. distribute-neg-frac 2.3%

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

    6. metadata-eval 2.3%

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

    7. add-sqr-sqrt 2.3%

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

    8. fabs-sqr 2.3%

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

    9. add-sqr-sqrt 2.3%

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

18. Applied egg-rr 2.3%

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

    1. distribute-frac-neg2 2.3%

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

    2. associate-*l/ 2.3%

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

    3. mul-1-neg 2.3%

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

    4. remove-double-neg 2.3%

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

20. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt{\pi}}} \]
21. Step-by-step derivation

    1. add-sqr-sqrt 2.3%

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

    2. sqrt-unprod 2.2%

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

    3. frac-times 2.2%

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

    4. pow2 2.2%

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

    5. add-exp-log 2.2%

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

    6. neg-log 2.2%

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

    7. add-sqr-sqrt 0.0%

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

    8. sqrt-unprod 50.5%

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

    9. sqr-neg 50.5%

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

    10. sqrt-unprod 50.5%

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

    11. add-sqr-sqrt 50.5%

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

    12. add-exp-log 50.5%

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

    13. add-sqr-sqrt 50.5%

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

22. Applied egg-rr 50.5%

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

Alternative 8: 5.4% accurate, 20.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (sqrt PI)))

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(x / sqrt(pi))
end

MATLAB

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

Wolfram

code[x_] := N[(x / N[Sqrt[Pi], $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%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 2.3%

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

14. Taylor expanded in x around inf 2.3%

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

    1. associate-*r/ 2.3%

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

    2. *-rgt-identity 2.3%

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

16. Simplified 2.3%

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

    1. frac-2neg 2.3%

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

    2. div-inv 2.3%

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

    3. sqrt-div 2.3%

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

    4. metadata-eval 2.3%

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

    5. distribute-neg-frac 2.3%

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

    6. metadata-eval 2.3%

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

    7. add-sqr-sqrt 2.3%

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

    8. fabs-sqr 2.3%

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

    9. add-sqr-sqrt 2.3%

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

18. Applied egg-rr 2.3%

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

    1. distribute-frac-neg2 2.3%

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

    2. associate-*l/ 2.3%

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

    3. mul-1-neg 2.3%

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

    4. remove-double-neg 2.3%

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

20. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt{\pi}}} \]
21. Step-by-step derivation

    1. div-inv 2.3%

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

    2. pow1/2 2.3%

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

    3. pow-flip 2.3%

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

    4. metadata-eval 2.3%

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

    5. expm1-log1p-u 2.3%

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

    6. add-exp-log 2.3%

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

    7. neg-log 2.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{-\log x}} \cdot {\pi}^{-0.5}\right)\right) \]

    8. add-exp-log 2.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{-\log x} \cdot \color{blue}{e^{\log \left({\pi}^{-0.5}\right)}}\right)\right) \]

    9. exp-sum 2.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\left(-\log x\right) + \log \left({\pi}^{-0.5}\right)}}\right)\right) \]

    10. expm1-undefine 1.7%

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

22. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1} \]
23. Step-by-step derivation

    1. log1p-undefine 5.4%

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

    2. rem-exp-log 5.4%

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

    3. associate-+r- 5.4%

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

    4. +-commutative 5.4%

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

    5. associate-+l- 5.4%

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

    6. metadata-eval 5.4%

       \[\leadsto \frac{x}{\sqrt{\pi}} - \color{blue}{0} \]

    7. --rgt-identity 5.4%

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

24. Simplified 5.4%

    \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}}} \]
25. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))))