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

Percentage Accurate: 100.0% → 100.0%

Time: 36.0s

Alternatives: 14

Speedup: 2.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{{x}^{-3}}\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{t\_0}^{2}}\right) + \log \left(\sqrt[3]{t\_0}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (pow x -3.0))))
   (*
    (/ (pow (exp x) x) (sqrt PI))
    (fma
     0.5
     (+ (log (cbrt (pow t_0 2.0))) (log (cbrt t_0)))
     (*
      (/ 1.0 (fabs x))
      (+ 1.0 (+ (* 1.875 (pow x -6.0)) (* 0.75 (pow x -4.0)))))))))

C

double code(double x) {
	double t_0 = exp(pow(x, -3.0));
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(0.5, (log(cbrt(pow(t_0, 2.0))) + log(cbrt(t_0))), ((1.0 / fabs(x)) * (1.0 + ((1.875 * pow(x, -6.0)) + (0.75 * pow(x, -4.0))))));
}

Julia

function code(x)
	t_0 = exp((x ^ -3.0))
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(0.5, Float64(log(cbrt((t_0 ^ 2.0))) + log(cbrt(t_0))), Float64(Float64(1.0 / abs(x)) * Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(0.75 * (x ^ -4.0)))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[Log[N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[t$95$0, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. add-cube-cbrt 100.0%

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

   3. log-prod 100.0%

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

6. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

   9. inv-pow 100.0%

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

   10. pow-pow 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

   14. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. add-log-exp 100.0%

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

    2. *-un-lft-identity 100.0%

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

    3. log-prod 100.0%

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

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{\left(e^{\color{blue}{0} + \log \left(e^{\frac{1}{{x}^{3}}}\right)}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{{x}^{-3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \]

    5. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{\left(e^{0 + \color{blue}{\frac{1}{{x}^{3}}}}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{{x}^{-3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \]

    6. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{\left(e^{0 + \color{blue}{{x}^{\left(-3\right)}}}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{{x}^{-3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{\left(e^{0 + {x}^{\color{blue}{-3}}}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{{x}^{-3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{{\left(e^{\color{blue}{0 + {x}^{-3}}}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{{x}^{-3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \]
12. Step-by-step derivation

    1. +-lft-identity 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt[3]{e^{{x}^{-3}}}\right)\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, t\_0 + 2 \cdot t\_0, \frac{1}{\left|x\right|} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + 0.75 \cdot {x}^{-4}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (cbrt (exp (pow x -3.0))))))
   (*
    (/ (pow (exp x) x) (sqrt PI))
    (fma
     0.5
     (+ t_0 (* 2.0 t_0))
     (*
      (/ 1.0 (fabs x))
      (+ 1.0 (+ (* 1.875 (pow x -6.0)) (* 0.75 (pow x -4.0)))))))))

C

double code(double x) {
	double t_0 = log(cbrt(exp(pow(x, -3.0))));
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(0.5, (t_0 + (2.0 * t_0)), ((1.0 / fabs(x)) * (1.0 + ((1.875 * pow(x, -6.0)) + (0.75 * pow(x, -4.0))))));
}

Julia

function code(x)
	t_0 = log(cbrt(exp((x ^ -3.0))))
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(0.5, Float64(t_0 + Float64(2.0 * t_0)), Float64(Float64(1.0 / abs(x)) * Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(0.75 * (x ^ -4.0)))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[Power[N[Exp[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(t$95$0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. add-cube-cbrt 100.0%

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

   3. log-prod 100.0%

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

6. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

   9. inv-pow 100.0%

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

   10. pow-pow 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. fabs-sqr 100.0%

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

   13. add-sqr-sqrt 100.0%

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

   14. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. add-sqr-sqrt 100.0%

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

    2. log-prod 100.0%

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

11. Applied egg-rr 100.0%

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

    1. count-2 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{\left|x\right|} \cdot \left(1 + \log \left(e^{\mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (cbrt (pow PI 1.5)))
  (fma
   0.5
   (pow x -3.0)
   (*
    (/ 1.0 (fabs x))
    (+ 1.0 (log (exp (fma 0.75 (pow x -4.0) (* 1.875 (pow x -6.0))))))))))

C

double code(double x) {
	return (pow(exp(x), x) / cbrt(pow(((double) M_PI), 1.5))) * fma(0.5, pow(x, -3.0), ((1.0 / fabs(x)) * (1.0 + log(exp(fma(0.75, pow(x, -4.0), (1.875 * pow(x, -6.0))))))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / cbrt((pi ^ 1.5))) * fma(0.5, (x ^ -3.0), Float64(Float64(1.0 / abs(x)) * Float64(1.0 + log(exp(fma(0.75, (x ^ -4.0), Float64(1.875 * (x ^ -6.0)))))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Log[N[Exp[N[(0.75 * N[Power[x, -4.0], $MachinePrecision] + N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \color{blue}{0} + \log \left(e^{\frac{1}{{\left(\left|x\right|\right)}^{3}}}\right), \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{3}}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-3\right)}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   8. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-3\right)}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + {\color{blue}{x}}^{\left(-3\right)}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, 0 + {x}^{\color{blue}{-3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \color{blue}{0 + {x}^{-3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. inv-pow 100.0%

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

   9. pow-pow 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. fabs-sqr 100.0%

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

   12. add-sqr-sqrt 100.0%

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

   13. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. add-cbrt-cube 100.0%

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

    2. pow1/3 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. pow1 100.0%

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

    5. pow1/2 100.0%

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

    6. pow-prod-up 100.0%

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

    7. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{\left|x\right|} \cdot \left(1 + \log \left(e^{\mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. unpow1/3 100.0%

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

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{\left|x\right|} \cdot \left(1 + \log \left(e^{\mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)\right)\right) \]

15. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{\left|x\right|} \cdot \left(1 + \log \left(e^{\mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)\right)\right) \]
16. Add Preprocessing

Alternative 4: 100.0% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (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))) (fabs x))))))

C

double code(double x) {
	return (pow(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))) / fabs(x))));
}

Julia

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $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[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. exp-prod 100.0%

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

14. Applied egg-rr 100.0%

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

    1. add-cbrt-cube 100.0%

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

    2. pow1/3 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. pow1 100.0%

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

    5. pow1/2 100.0%

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

    6. pow-prod-up 100.0%

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

    7. metadata-eval 100.0%

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

16. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

    1. exp-prod 100.0%

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

14. Applied egg-rr 100.0%

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

15. Final simplification 100.0%

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

Alternative 6: 100.0% accurate, 2.9× speedup

Math

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

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))) (fabs 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))) / fabs(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(0.5 / Float64(x * x))) / abs(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[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 7: 99.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

    1. pow2 99.4%

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

    2. pow-exp 99.4%

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

    3. add-sqr-sqrt 99.4%

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

    4. pow-unpow 99.4%

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

15. Applied egg-rr 99.4%

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

16. Final simplification 99.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{{\left({\left(e^{x}\right)}^{\left(\sqrt{x}\right)}\right)}^{\left(\sqrt{x}\right)}}{\left|x\right|} \]
17. Add Preprocessing

Alternative 8: 99.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((sqrt(exp(x)) ^ (x * 2.0)) / abs(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

    1. pow2 99.4%

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

    2. pow-exp 99.4%

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

    3. add-sqr-sqrt 99.4%

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

    4. unpow-prod-down 99.4%

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

15. Applied egg-rr 99.4%

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

    1. pow-sqr 99.4%

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

    2. *-commutative 99.4%

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

17. Simplified 99.4%

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

18. Final simplification 99.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{{\left(\sqrt{e^{x}}\right)}^{\left(x \cdot 2\right)}}{\left|x\right|} \]
19. Add Preprocessing

Alternative 9: 99.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (exp((x ^ 2.0)) / abs(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

14. Final simplification 99.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} \]
15. Add Preprocessing

Alternative 10: 99.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

    1. pow2 99.4%

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

    2. pow-exp 99.4%

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

15. Applied egg-rr 99.4%

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

16. Final simplification 99.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} \]
17. Add Preprocessing

Alternative 11: 51.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right|} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

14. Taylor expanded in x around 0 51.3%

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

    1. +-commutative 51.3%

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

    2. unpow2 51.3%

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

    3. fma-define 51.3%

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

16. Simplified 51.3%

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

17. Final simplification 51.3%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right|} \]
18. Add Preprocessing

Alternative 12: 2.3% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 99.4%

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

14. Taylor expanded in x around 0 2.3%

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

15. Final simplification 2.3%

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

Alternative 13: 1.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{{x}^{-14}}{\pi} \cdot 3.515625} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (/ (pow x -14.0) PI) 3.515625)))

C

double code(double x) {
	return sqrt(((pow(x, -14.0) / ((double) M_PI)) * 3.515625));
}

Java

public static double code(double x) {
	return Math.sqrt(((Math.pow(x, -14.0) / Math.PI) * 3.515625));
}

Python

def code(x):
	return math.sqrt(((math.pow(x, -14.0) / math.pi) * 3.515625))

Julia

function code(x)
	return sqrt(Float64(Float64((x ^ -14.0) / pi) * 3.515625))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((((x ^ -14.0) / pi) * 3.515625));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Power[x, -14.0], $MachinePrecision] / Pi), $MachinePrecision] * 3.515625), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 1.7%

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

    1. *-commutative 1.7%

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

    2. associate-*r/ 1.7%

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

    3. *-rgt-identity 1.7%

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

15. Simplified 1.7%

    \[\leadsto \color{blue}{1.875 \cdot \frac{\sqrt{\frac{1}{\pi}}}{{x}^{7}}} \]
16. Step-by-step derivation

    1. add-sqr-sqrt 1.7%

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

    2. sqrt-unprod 1.7%

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

    3. *-commutative 1.7%

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

    4. *-commutative 1.7%

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

    5. swap-sqr 1.7%

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

17. Applied egg-rr 1.7%

    \[\leadsto \color{blue}{\sqrt{\left(\frac{1}{\pi} \cdot {x}^{-14}\right) \cdot 3.515625}} \]
18. Step-by-step derivation

    1. associate-*l* 1.7%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\pi} \cdot \left({x}^{-14} \cdot 3.515625\right)}} \]

    2. associate-*l/ 1.7%

       \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{-14} \cdot 3.515625\right)}{\pi}}} \]

    3. *-lft-identity 1.7%

       \[\leadsto \sqrt{\frac{\color{blue}{{x}^{-14} \cdot 3.515625}}{\pi}} \]

    4. associate-*l/ 1.7%

       \[\leadsto \sqrt{\color{blue}{\frac{{x}^{-14}}{\pi} \cdot 3.515625}} \]

19. Simplified 1.7%

    \[\leadsto \color{blue}{\sqrt{\frac{{x}^{-14}}{\pi} \cdot 3.515625}} \]

20. Final simplification 1.7%

    \[\leadsto \sqrt{\frac{{x}^{-14}}{\pi} \cdot 3.515625} \]
21. Add Preprocessing

Alternative 14: 1.7% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.875 * N[(N[Power[x, -7.0], $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%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. inv-pow 100.0%

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

   7. pow-pow 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

   11. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. +-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 1.7%

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

    1. *-commutative 1.7%

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

    2. associate-*r/ 1.7%

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

    3. *-rgt-identity 1.7%

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

15. Simplified 1.7%

    \[\leadsto \color{blue}{1.875 \cdot \frac{\sqrt{\frac{1}{\pi}}}{{x}^{7}}} \]
16. Step-by-step derivation

    1. *-un-lft-identity 1.7%

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

    2. *-un-lft-identity 1.7%

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

    3. associate-*l/ 1.7%

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

    4. pow-flip 1.7%

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

    5. metadata-eval 1.7%

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

    6. sqrt-div 1.7%

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

    7. metadata-eval 1.7%

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

    8. un-div-inv 1.7%

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

17. Applied egg-rr 1.7%

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

    1. *-lft-identity 1.7%

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

19. Simplified 1.7%

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

20. Final simplification 1.7%

    \[\leadsto 1.875 \cdot \frac{{x}^{-7}}{\sqrt{\pi}} \]
21. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))))