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

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 11

Speedup: 1.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x \cdot x}}{\left|x\right|}\\ t_1 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(t\_1 + \mathsf{fma}\left(0.5, t\_0, 0.75 \cdot \frac{\frac{t\_0}{\left|x\right|}}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot t\_1\right) \cdot t\_1\right) \cdot t\_1\right) \cdot t\_1\right) \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 (* x x)) (fabs x))) (t_1 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ t_1 (fma 0.5 t_0 (* 0.75 (/ (/ t_0 (fabs x)) (fabs x)))))
     (* (/ 15.0 8.0) (* (* (* (* (* (* t_1 t_1) t_1) t_1) t_1) t_1) t_1))))))

C

double code(double x) {
	double t_0 = (1.0 / (x * x)) / fabs(x);
	double t_1 = 1.0 / fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * ((t_1 + fma(0.5, t_0, (0.75 * ((t_0 / fabs(x)) / fabs(x))))) + ((15.0 / 8.0) * ((((((t_1 * t_1) * t_1) * t_1) * t_1) * t_1) * t_1)));
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(x * x)) / abs(x))
	t_1 = Float64(1.0 / abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(t_1 + fma(0.5, t_0, Float64(0.75 * Float64(Float64(t_0 / abs(x)) / abs(x))))) + Float64(Float64(15.0 / 8.0) * Float64(Float64(Float64(Float64(Float64(Float64(t_1 * t_1) * t_1) * t_1) * t_1) * t_1) * t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $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[(t$95$1 + N[(0.5 * t$95$0 + N[(0.75 * N[(N[(t$95$0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]

4. Applied rewrites 100.0%

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

Alternative 3: 99.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\pi}} \]

   2. +-commutative N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   3. pow-flip N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{15}{8} \cdot {\left(\left|x\right|\right)}^{\left(\mathsf{neg}\left(7\right)\right)} + \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{15}{8} \cdot {\left(\left|x\right|\right)}^{-7} + \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   5. *-commutative N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, {\left(\left|x\right|\right)}^{-7} \cdot \frac{15}{8} + \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   6. lower-fma.f64 N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)\right)}{\sqrt{\pi}} \]

   7. lift-pow.f64 N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)\right)}{\sqrt{\pi}} \]

   8. lift-fabs.f64 N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)\right)}{\sqrt{\pi}} \]

   9. metadata-eval N/A

      \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)\right)}{\sqrt{\pi}} \]

   10. lift-/.f64 N/A

       \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)\right)}{\sqrt{\pi}} \]

   11. lift-fabs.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. associate-/l* N/A

      \[\leadsto e^{{\left(\left|x\right|\right)}^{2}} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}}{\sqrt{\mathsf{PI}\left(\right)}}} \]

6. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{x \cdot x}}{x \cdot x} \cdot \frac{\frac{0.75}{\left(x \cdot x\right) \cdot \left|x\right|} + \frac{0.5}{\left|x\right|}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (*
    (/ (exp (* x x)) (* x x))
    (/ (+ (/ 0.75 (* (* x x) (fabs x))) (/ 0.5 (fabs x))) (sqrt PI)))
   (*
    (fma x x 1.0)
    (/ (fma (pow (fabs x) -7.0) 1.875 (/ 1.0 (fabs x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (exp((x * x)) / (x * x)) * (((0.75 / ((x * x) * fabs(x))) + (0.5 / fabs(x))) / sqrt(((double) M_PI)));
	} else {
		tmp = fma(x, x, 1.0) * (fma(pow(fabs(x), -7.0), 1.875, (1.0 / fabs(x))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(x * x)) / Float64(x * x)) * Float64(Float64(Float64(0.75 / Float64(Float64(x * x) * abs(x))) + Float64(0.5 / abs(x))) / sqrt(pi)));
	else
		tmp = Float64(fma(x, x, 1.0) * Float64(fma((abs(x) ^ -7.0), 1.875, Float64(1.0 / abs(x))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   6. Applied rewrites 48.9%

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

   8. Applied rewrites 48.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      2. associate-/l* N/A

         \[\leadsto e^{{\left(\left|x\right|\right)}^{2}} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2}\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \color{blue}{\frac{15}{8}}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \left(x \cdot x + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      3. lower-fma.f64 52.6

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

   9. Applied rewrites 52.6%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{0.75}{\left(x \cdot x\right) \cdot \left|x\right|} + \frac{0.5}{\left|x\right|}\right) \cdot e^{x \cdot x}}{\left(x \cdot x\right) \cdot \sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+154)
   (/
    (* (+ (/ 0.75 (* (* x x) (fabs x))) (/ 0.5 (fabs x))) (exp (* x x)))
    (* (* x x) (sqrt PI)))
   (*
    (fma x x 1.0)
    (/ (fma (pow (fabs x) -7.0) 1.875 (/ 1.0 (fabs x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+154) {
		tmp = (((0.75 / ((x * x) * fabs(x))) + (0.5 / fabs(x))) * exp((x * x))) / ((x * x) * sqrt(((double) M_PI)));
	} else {
		tmp = fma(x, x, 1.0) * (fma(pow(fabs(x), -7.0), 1.875, (1.0 / fabs(x))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+154)
		tmp = Float64(Float64(Float64(Float64(0.75 / Float64(Float64(x * x) * abs(x))) + Float64(0.5 / abs(x))) * exp(Float64(x * x))) / Float64(Float64(x * x) * sqrt(pi)));
	else
		tmp = Float64(fma(x, x, 1.0) * Float64(fma((abs(x) ^ -7.0), 1.875, Float64(1.0 / abs(x))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+154], N[(N[(N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   6. Applied rewrites 48.9%

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

   if 1.00000000000000004e154 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      2. associate-/l* N/A

         \[\leadsto e^{{\left(\left|x\right|\right)}^{2}} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2}\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \color{blue}{\frac{15}{8}}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \left(x \cdot x + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      3. lower-fma.f64 52.6

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

   9. Applied rewrites 52.6%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+154}:\\ \;\;\;\;\frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot 0.5}{\left(x \cdot x\right) \cdot \sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+154)
   (/ (* (/ (exp (* x x)) (fabs x)) 0.5) (* (* x x) (sqrt PI)))
   (*
    (fma x x 1.0)
    (/ (fma (pow (fabs x) -7.0) 1.875 (/ 1.0 (fabs x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+154) {
		tmp = ((exp((x * x)) / fabs(x)) * 0.5) / ((x * x) * sqrt(((double) M_PI)));
	} else {
		tmp = fma(x, x, 1.0) * (fma(pow(fabs(x), -7.0), 1.875, (1.0 / fabs(x))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+154)
		tmp = Float64(Float64(Float64(exp(Float64(x * x)) / abs(x)) * 0.5) / Float64(Float64(x * x) * sqrt(pi)));
	else
		tmp = Float64(fma(x, x, 1.0) * Float64(fma((abs(x) ^ -7.0), 1.875, Float64(1.0 / abs(x))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+154], N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   6. Applied rewrites 48.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. metadata-eval 48.9

         \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot 0.5}{\left(x \cdot x\right) \cdot \sqrt{\pi}} \]

   9. Applied rewrites 48.9%

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

   if 1.00000000000000004e154 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      2. associate-/l* N/A

         \[\leadsto e^{{\left(\left|x\right|\right)}^{2}} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2}\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \color{blue}{\frac{15}{8}}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \left(x \cdot x + 1\right) \cdot \frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

      3. lower-fma.f64 52.6

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

   9. Applied rewrites 52.6%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 48.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

6. Applied rewrites 48.9%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. metadata-eval 48.9

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot 0.5}{\left(x \cdot x\right) \cdot \sqrt{\pi}} \]

9. Applied rewrites 48.9%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot 0.5}{\color{blue}{\left(x \cdot x\right)} \cdot \sqrt{\pi}} \]
10. Add Preprocessing

Alternative 9: 32.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \mathsf{fma}\left(0.5, \frac{\frac{1}{x \cdot x}}{\left|x\right|}, 0.75 \cdot \frac{\frac{\frac{\frac{1}{x \cdot x}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right)\right)} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

6. Applied rewrites 48.9%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

9. Applied rewrites 32.7%

   \[\leadsto \frac{e^{x \cdot x}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(x \cdot x\right)} \cdot \color{blue}{0.5} \]
10. Add Preprocessing

Alternative 10: 1.8% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   7. lift-fabs.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   8. lift-PI.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 1.8

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

7. Applied rewrites 1.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(x \cdot x\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(x \cdot x\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   8. associate-*l* N/A

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

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left({x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   10. lower-*.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   14. lift-PI.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-*.f64 1.8

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

9. Applied rewrites 1.8%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    8. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

    9. mult-flip-rev N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \frac{1}{\left|x\right|}}{\left(x \cdot x\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

    10. pow2 N/A

        \[\leadsto \frac{\frac{1}{2} \cdot \frac{1}{\left|x\right|}}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

    11. lower-/.f64 N/A

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

    12. mult-flip-rev N/A

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

    13. lower-/.f64 N/A

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

    14. lift-fabs.f64 N/A

        \[\leadsto \frac{\frac{\frac{1}{2}}{\left|x\right|}}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

    15. pow2 N/A

        \[\leadsto \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

    16. lift-PI.f64 N/A

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

    17. lift-sqrt.f64 N/A

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

    18. lift-*.f64 N/A

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

    19. lift-*.f64 1.8

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

11. Applied rewrites 1.8%

    \[\leadsto \frac{\frac{0.5}{\left|x\right|}}{\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 11: 1.8% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   7. lift-fabs.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   8. lift-PI.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 1.8

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

7. Applied rewrites 1.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(x \cdot x\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(x \cdot x\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot {x}^{2}} \]

   8. associate-*l* N/A

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

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left({x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   10. lower-*.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   14. lift-PI.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-*.f64 1.8

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

9. Applied rewrites 1.8%

   \[\leadsto \frac{0.5}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\pi}}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(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)))))))