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

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 6

Speedup: N/A×

• 99.6% 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, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (*
    (exp (* x x))
    (fma
     0.75
     (pow x -5.0)
     (/ (+ 1.0 (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))) x)))
   (pow PI 0.25))
  (pow PI 0.25)))

C

double code(double x) {
	return ((exp((x * x)) * fma(0.75, pow(x, -5.0), ((1.0 + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))) / x))) / pow(((double) M_PI), 0.25)) / pow(((double) M_PI), 0.25);
}

Julia

function code(x)
	return Float64(Float64(Float64(exp(Float64(x * x)) * fma(0.75, (x ^ -5.0), Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0)))) / x))) / (pi ^ 0.25)) / (pi ^ 0.25))
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.25], $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. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \color{blue}{\frac{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)}{x}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

9. Simplified 100.0%

   \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \color{blue}{\frac{1 + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)}{x}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
12. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

13. Applied egg-rr 100.0%

    \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
14. Add Preprocessing

Alternative 2: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 48.5%

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

   1. associate-*r* 48.5%

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

   2. *-commutative 48.5%

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

   3. distribute-rgt-out 48.5%

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

   4. unpow1/2 48.5%

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

   5. rem-exp-log 48.5%

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

   6. exp-neg 48.5%

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

   7. exp-prod 48.5%

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

   8. distribute-lft-neg-out 48.5%

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

   9. distribute-rgt-neg-in 48.5%

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

   10. metadata-eval 48.5%

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

   11. exp-to-pow 48.5%

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

   12. fma-define 48.5%

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

9. Simplified 48.5%

   \[\leadsto \color{blue}{\frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, \frac{e^{{x}^{2}}}{{x}^{2}}, e^{{x}^{2}}\right)}{x}} \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

11. Applied egg-rr 48.5%

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

12. Taylor expanded in x around inf 48.5%

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

    1. associate-*r/ 48.5%

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

    2. associate-*l/ 48.5%

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

    3. metadata-eval 48.5%

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

    4. associate-*r/ 48.5%

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

    5. *-lft-identity 48.5%

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

    6. distribute-rgt-out 99.7%

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

    7. associate-*r/ 99.7%

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

    8. metadata-eval 99.7%

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

14. Simplified 99.7%

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

Alternative 3: 99.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, {x}^{-2}, e^{x \cdot x}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (pow PI -0.5) (fma 0.5 (pow x -2.0) (exp (* x x)))) x))

C

double code(double x) {
	return (pow(((double) M_PI), -0.5) * fma(0.5, pow(x, -2.0), exp((x * x)))) / x;
}

Julia

function code(x)
	return Float64(Float64((pi ^ -0.5) * fma(0.5, (x ^ -2.0), exp(Float64(x * x)))) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 48.5%

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

   1. associate-*r* 48.5%

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

   2. *-commutative 48.5%

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

   3. distribute-rgt-out 48.5%

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

   4. unpow1/2 48.5%

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

   5. rem-exp-log 48.5%

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

   6. exp-neg 48.5%

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

   7. exp-prod 48.5%

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

   8. distribute-lft-neg-out 48.5%

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

   9. distribute-rgt-neg-in 48.5%

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

   10. metadata-eval 48.5%

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

   11. exp-to-pow 48.5%

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

   12. fma-define 48.5%

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

9. Simplified 48.5%

   \[\leadsto \color{blue}{\frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, \frac{e^{{x}^{2}}}{{x}^{2}}, e^{{x}^{2}}\right)}{x}} \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

11. Applied egg-rr 48.5%

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

12. Taylor expanded in x around 0 99.7%

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

    1. exp-to-pow 99.7%

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

    2. *-commutative 99.7%

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

    3. rec-exp 99.7%

       \[\leadsto \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, \color{blue}{e^{-2 \cdot \log x}}, e^{x \cdot x}\right)}{x} \]

    4. *-commutative 99.7%

       \[\leadsto \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, e^{-\color{blue}{\log x \cdot 2}}, e^{x \cdot x}\right)}{x} \]

    5. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, e^{\color{blue}{\log x \cdot \left(-2\right)}}, e^{x \cdot x}\right)}{x} \]

    6. metadata-eval 99.7%

       \[\leadsto \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, e^{\log x \cdot \color{blue}{-2}}, e^{x \cdot x}\right)}{x} \]

    7. exp-to-pow 99.7%

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

14. Simplified 99.7%

    \[\leadsto \frac{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, \color{blue}{{x}^{-2}}, e^{x \cdot x}\right)}{x} \]
15. Add Preprocessing

Alternative 4: 99.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ e^{\sqrt[3]{{x}^{6}}} \cdot \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (cbrt (pow x 6.0))) (/ (pow PI -0.5) x)))

C

double code(double x) {
	return exp(cbrt(pow(x, 6.0))) * (pow(((double) M_PI), -0.5) / x);
}

Java

public static double code(double x) {
	return Math.exp(Math.cbrt(Math.pow(x, 6.0))) * (Math.pow(Math.PI, -0.5) / x);
}

Julia

function code(x)
	return Float64(exp(cbrt((x ^ 6.0))) * Float64((pi ^ -0.5) / x))
end

Wolfram

code[x_] := N[(N[Exp[N[Power[N[Power[x, 6.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

   3. unpow1/2 99.7%

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

   4. rem-exp-log 99.7%

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

   5. exp-neg 99.7%

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

   6. exp-prod 99.7%

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

   7. distribute-lft-neg-out 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\color{blue}{-\log \pi \cdot 0.5}}}{x} \]

   8. distribute-rgt-neg-in 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\log \pi \cdot \color{blue}{-0.5}}}{x} \]

   10. exp-to-pow 99.7%

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

9. Simplified 99.7%

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

    1. add-cbrt-cube 99.7%

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

    2. pow3 99.7%

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

    3. unpow2 99.7%

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

    4. pow-prod-down 99.7%

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

    5. pow-prod-up 99.7%

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

    6. metadata-eval 99.7%

       \[\leadsto e^{\sqrt[3]{{x}^{\color{blue}{6}}}} \cdot \frac{{\pi}^{-0.5}}{x} \]

11. Applied egg-rr 99.7%

    \[\leadsto e^{\color{blue}{\sqrt[3]{{x}^{6}}}} \cdot \frac{{\pi}^{-0.5}}{x} \]
12. Add Preprocessing

Alternative 5: 99.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

   3. unpow1/2 99.7%

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

   4. rem-exp-log 99.7%

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

   5. exp-neg 99.7%

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

   6. exp-prod 99.7%

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

   7. distribute-lft-neg-out 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\color{blue}{-\log \pi \cdot 0.5}}}{x} \]

   8. distribute-rgt-neg-in 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\log \pi \cdot \color{blue}{-0.5}}}{x} \]

   10. exp-to-pow 99.7%

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

9. Simplified 99.7%

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

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \frac{1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

11. Applied egg-rr 99.7%

    \[\leadsto e^{\color{blue}{x \cdot x}} \cdot \frac{{\pi}^{-0.5}}{x} \]
12. Add Preprocessing

Alternative 6: 2.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $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. associate-*l/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. associate-/r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

7. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

   3. unpow1/2 99.7%

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

   4. rem-exp-log 99.7%

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

   5. exp-neg 99.7%

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

   6. exp-prod 99.7%

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

   7. distribute-lft-neg-out 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\color{blue}{-\log \pi \cdot 0.5}}}{x} \]

   8. distribute-rgt-neg-in 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto e^{{x}^{2}} \cdot \frac{e^{\log \pi \cdot \color{blue}{-0.5}}}{x} \]

   10. exp-to-pow 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 2.3%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}} \]
11. 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%

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

    3. rem-exp-log 2.3%

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

    4. exp-neg 2.3%

       \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \pi}}}}{x} \]

    5. unpow1/2 2.3%

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

    6. exp-prod 2.3%

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

    7. distribute-lft-neg-out 2.3%

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

    8. distribute-rgt-neg-in 2.3%

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

    9. metadata-eval 2.3%

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

    10. exp-to-pow 2.3%

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

12. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))))))