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

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 6

Speedup: 4.0×

• 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, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-out 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

11. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-out 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. pow2 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-out 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.7%

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

    1. metadata-eval 99.7%

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

    2. pow-prod-up 99.7%

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

    3. inv-pow 99.7%

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

    4. inv-pow 99.7%

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

    5. frac-2neg 99.7%

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

    6. metadata-eval 99.7%

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

    7. frac-times 99.7%

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

    8. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-out 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.7%

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

10. Taylor expanded in x around 0 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{1}{x} + \color{blue}{\frac{1.875}{{x}^{7}}}\right) \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-out 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.7%

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

10. Taylor expanded in x around inf 99.7%

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

Alternative 6: 1.8% accurate, 19.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (0.5 / (x * (sqrt(((double) M_PI)) * x))) / x;
}

Java

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

Python

def code(x):
	return (0.5 / (x * (math.sqrt(math.pi) * x))) / x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 / N[(x * N[(N[Sqrt[Pi], $MachinePrecision] * x), $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. Taylor expanded in x around 0 1.8%

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

   1. associate-*r* 1.8%

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

   2. *-commutative 1.8%

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

   3. associate-*r/ 1.8%

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

   4. metadata-eval 1.8%

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

   5. unpow2 1.8%

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

   6. sqr-abs 1.8%

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

   7. unpow3 1.8%

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

7. Simplified 1.8%

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

   1. *-commutative 1.8%

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

   2. associate-*l/ 1.8%

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

   3. cube-mult 1.8%

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

   4. sqr-abs 1.8%

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

   5. pow2 1.8%

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

   6. associate-*l/ 1.8%

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

   7. sqrt-div 1.8%

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

   8. metadata-eval 1.8%

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

   9. un-div-inv 1.8%

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

9. Applied egg-rr 1.8%

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

    1. associate-/l* 1.8%

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

11. Simplified 1.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}}} \]
12. Step-by-step derivation

    1. *-commutative 1.8%

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

    2. associate-*l/ 1.8%

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

    3. metadata-eval 1.8%

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

    4. pow-flip 1.8%

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

    5. associate-/r/ 1.8%

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

    6. clear-num 1.8%

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

    7. cube-mult 1.8%

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

    8. pow2 1.8%

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

    9. associate-/r* 1.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.5}{x}}{{x}^{2}}}}{\sqrt{\pi}} \]

    10. associate-/r* 1.8%

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

    11. associate-/l/ 1.8%

        \[\leadsto \color{blue}{\frac{\frac{\frac{0.5}{x}}{\sqrt{\pi}}}{{x}^{2}}} \]

    12. pow2 1.8%

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

    13. associate-/r* 1.8%

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

13. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.5}{x}}{\sqrt{\pi}}}{x}}{x}} \]
14. Step-by-step derivation

    1. *-un-lft-identity 1.8%

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

    2. associate-/l/ 1.8%

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

    3. *-commutative 1.8%

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

15. Applied egg-rr 1.8%

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{0.5}{x}}{\sqrt{\pi} \cdot x}}}{x} \]
16. Step-by-step derivation

    1. *-lft-identity 1.8%

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

    2. associate-/l/ 1.8%

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

    3. *-commutative 1.8%

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

17. Simplified 1.8%

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

18. Final simplification 1.8%

    \[\leadsto \frac{\frac{0.5}{x \cdot \left(\sqrt{\pi} \cdot x\right)}}{x} \]
19. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))))