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

Percentage Accurate: 100.0% → 100.0%

Time: 23.6s

Alternatives: 13

Speedup: 3.3×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(Math.exp(x)), (x * 2.0)) * (Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + ((((1.0 + (0.75 * Math.pow(x, -5.0))) + -1.0) + (1.875 / Math.pow(x, 7.0))) + (1.0 / x))));
}

Python

def code(x):
	return math.pow(math.sqrt(math.exp(x)), (x * 2.0)) * (math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + ((((1.0 + (0.75 * math.pow(x, -5.0))) + -1.0) + (1.875 / math.pow(x, 7.0))) + (1.0 / x))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (sqrt(exp(x)) ^ (x * 2.0)) * (sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + ((((1.0 + (0.75 * (x ^ -5.0))) + -1.0) + (1.875 / (x ^ 7.0))) + (1.0 / x))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   2. expm1-undefine 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(\color{blue}{0.75 \cdot \frac{1}{{x}^{5}}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   4. pow-flip 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot \color{blue}{{x}^{\left(-5\right)}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   5. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{\color{blue}{-5}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]
10. Step-by-step derivation

    1. add-sqr-sqrt 100.0%

       \[\leadsto {\color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

    2. unpow-prod-down 100.0%

       \[\leadsto \color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{x} \cdot {\left(\sqrt{e^{x}}\right)}^{x}\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{x} \cdot {\left(\sqrt{e^{x}}\right)}^{x}\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]
12. Step-by-step derivation

    1. pow-sqr 100.0%

       \[\leadsto \color{blue}{{\left(\sqrt{e^{x}}\right)}^{\left(2 \cdot x\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

    2. *-commutative 100.0%

       \[\leadsto {\left(\sqrt{e^{x}}\right)}^{\color{blue}{\left(x \cdot 2\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

13. Simplified 100.0%

    \[\leadsto \color{blue}{{\left(\sqrt{e^{x}}\right)}^{\left(x \cdot 2\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]
14. Step-by-step derivation

    1. sub-neg 100.0%

       \[\leadsto {\left(\sqrt{e^{x}}\right)}^{\left(x \cdot 2\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} + \left(-1\right)\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

    2. log1p-undefine 100.0%

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

    3. rem-exp-log 100.0%

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

    4. metadata-eval 100.0%

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

15. Applied egg-rr 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   2. expm1-undefine 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(\color{blue}{0.75 \cdot \frac{1}{{x}^{5}}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   4. pow-flip 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot \color{blue}{{x}^{\left(-5\right)}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   5. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{\color{blue}{-5}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]
10. Step-by-step derivation

    1. sub-neg 100.0%

       \[\leadsto {\left(\sqrt{e^{x}}\right)}^{\left(x \cdot 2\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} + \left(-1\right)\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

    2. log1p-undefine 100.0%

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

    3. rem-exp-log 100.0%

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

    4. metadata-eval 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   2. expm1-undefine 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(\color{blue}{0.75 \cdot \frac{1}{{x}^{5}}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   4. pow-flip 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot \color{blue}{{x}^{\left(-5\right)}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

   5. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{\color{blue}{-5}}\right)} - 1\right) + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right)} + \frac{1.875}{{x}^{7}}\right) + \frac{1}{x}\right)\right)\right) \]

10. Taylor expanded in x around 0 100.0%

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

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left({\pi}^{-0.5} \cdot \left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right) \]
14. Add Preprocessing

Alternative 4: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. associate-+r+ 100.0%

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

    2. +-commutative 100.0%

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

    3. associate-+r+ 100.0%

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

    4. +-commutative 100.0%

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

    5. associate-+l+ 100.0%

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

    6. associate-*r/ 100.0%

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

    7. metadata-eval 100.0%

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

    8. +-commutative 100.0%

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

    9. associate-*r/ 100.0%

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

    10. metadata-eval 100.0%

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

    11. associate-*r/ 100.0%

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

    12. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 5: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ e^{{x}^{2}} \cdot \frac{\frac{1}{x} + \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.exp(Math.pow(x, 2.0)) * (((1.0 / x) + ((0.5 / Math.pow(x, 3.0)) + (0.75 / Math.pow(x, 5.0)))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return math.exp(math.pow(x, 2.0)) * (((1.0 / x) + ((0.5 / math.pow(x, 3.0)) + (0.75 / math.pow(x, 5.0)))) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(exp((x ^ 2.0)) * Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.5 / (x ^ 3.0)) + Float64(0.75 / (x ^ 5.0)))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = exp((x ^ 2.0)) * (((1.0 / x) + ((0.5 / (x ^ 3.0)) + (0.75 / (x ^ 5.0)))) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.6%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\color{blue}{0.5 \cdot \frac{1}{{x}^{3}} + \left(0.75 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right)}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. +-commutative 99.6%

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

    2. +-commutative 99.6%

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

    3. associate-+l+ 99.6%

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

    4. associate-*r/ 99.6%

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

    5. metadata-eval 99.6%

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

    6. associate-*r/ 99.6%

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

    7. metadata-eval 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in x around inf 99.6%

    \[\leadsto \color{blue}{e^{{x}^{2}}} \cdot \frac{\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)}{\sqrt{\pi}} \]

13. Final simplification 99.6%

    \[\leadsto e^{{x}^{2}} \cdot \frac{\frac{1}{x} + \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 6: 99.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{{x}^{2}} \cdot \frac{\frac{0.5}{{x}^{3}} + \frac{1}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

    1. associate-*r/ 99.5%

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

    2. metadata-eval 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in x around inf 99.5%

    \[\leadsto \color{blue}{e^{{x}^{2}}} \cdot \frac{\frac{1}{x} + \frac{0.5}{{x}^{3}}}{\sqrt{\pi}} \]

13. Final simplification 99.5%

    \[\leadsto e^{{x}^{2}} \cdot \frac{\frac{0.5}{{x}^{3}} + \frac{1}{x}}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 7: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

    1. associate-*r/ 99.5%

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

    2. pow-exp 99.5%

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

    3. pow2 99.5%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \frac{\frac{1}{x} \cdot e^{{x}^{2}}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 8: 99.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

    1. add0 99.5%

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

    2. *-commutative 99.5%

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

    3. associate-/l/ 99.5%

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

    4. pow-exp 99.5%

       \[\leadsto \frac{1}{\sqrt{\pi} \cdot x} \cdot \color{blue}{e^{x \cdot x}} + 0 \]

    5. pow2 99.5%

       \[\leadsto \frac{1}{\sqrt{\pi} \cdot x} \cdot e^{\color{blue}{{x}^{2}}} + 0 \]

11. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi} \cdot x} \cdot e^{{x}^{2}} + 0} \]
12. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \color{blue}{\frac{1 \cdot e^{{x}^{2}}}{\sqrt{\pi} \cdot x}} + 0 \]

    2. add0 99.5%

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

    3. *-lft-identity 99.5%

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

    4. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 9: 69.0% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

10. Taylor expanded in x around 0 72.0%

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

    1. associate-+r+ 72.0%

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

    2. +-commutative 72.0%

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

    3. *-commutative 72.0%

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

    4. associate-*r* 72.0%

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

    5. distribute-rgt-out 72.0%

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

    6. distribute-lft-out 72.0%

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

    7. unpow-1 72.0%

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

    8. metadata-eval 72.0%

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

    9. pow-sqr 72.0%

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

    10. rem-sqrt-square 72.0%

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

    11. sqr-pow 72.0%

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

    12. fabs-sqr 72.0%

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

    13. sqr-pow 72.0%

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

12. Simplified 72.0%

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

13. Final simplification 72.0%

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

Alternative 10: 69.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

10. Taylor expanded in x around 0 72.0%

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

    1. associate-+r+ 72.0%

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

    2. +-commutative 72.0%

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

    3. *-commutative 72.0%

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

    4. associate-*r* 72.0%

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

    5. distribute-rgt-out 72.0%

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

    6. distribute-lft-out 72.0%

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

    7. unpow-1 72.0%

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

    8. metadata-eval 72.0%

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

    9. pow-sqr 72.0%

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

    10. rem-sqrt-square 72.0%

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

    11. sqr-pow 72.0%

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

    12. fabs-sqr 72.0%

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

    13. sqr-pow 72.0%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around inf 72.0%

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

14. Final simplification 72.0%

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

Alternative 11: 69.0% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

10. Taylor expanded in x around 0 72.0%

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

    1. associate-+r+ 72.0%

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

    2. +-commutative 72.0%

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

    3. *-commutative 72.0%

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

    4. associate-*r* 72.0%

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

    5. distribute-rgt-out 72.0%

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

    6. distribute-lft-out 72.0%

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

    7. unpow-1 72.0%

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

    8. metadata-eval 72.0%

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

    9. pow-sqr 72.0%

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

    10. rem-sqrt-square 72.0%

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

    11. sqr-pow 72.0%

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

    12. fabs-sqr 72.0%

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

    13. sqr-pow 72.0%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around inf 72.0%

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

    1. associate-*r* 72.0%

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

15. Simplified 72.0%

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

16. Final simplification 72.0%

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

Alternative 12: 5.4% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

10. Taylor expanded in x around 0 5.5%

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

    1. distribute-rgt-out 5.5%

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

    2. unpow-1 5.5%

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

    3. metadata-eval 5.5%

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

    4. pow-sqr 5.5%

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

    5. rem-sqrt-square 5.5%

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

    6. sqr-pow 5.5%

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

    7. fabs-sqr 5.5%

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

    8. sqr-pow 5.5%

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

12. Simplified 5.5%

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

13. Final simplification 5.5%

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

Alternative 13: 2.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add0 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add0 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.5%

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

10. Taylor expanded in x around 0 72.0%

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

    1. associate-+r+ 72.0%

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

    2. +-commutative 72.0%

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

    3. *-commutative 72.0%

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

    4. associate-*r* 72.0%

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

    5. distribute-rgt-out 72.0%

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

    6. distribute-lft-out 72.0%

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

    7. unpow-1 72.0%

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

    8. metadata-eval 72.0%

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

    9. pow-sqr 72.0%

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

    10. rem-sqrt-square 72.0%

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

    11. sqr-pow 72.0%

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

    12. fabs-sqr 72.0%

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

    13. sqr-pow 72.0%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around 0 2.3%

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

    1. associate-*l/ 2.3%

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

    2. *-lft-identity 2.3%

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

15. Simplified 2.3%

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

16. Final simplification 2.3%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{x} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))))