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

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 10

Speedup: 1.9×

• 99.4% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \frac{\left(\frac{0.5}{x \cdot x} - -1\right) + \left(\frac{0.75}{t\_0 \cdot x} + \frac{1.875}{t\_0 \cdot t\_0}\right)}{\sqrt{x} \cdot \sqrt{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp (- x)) (- x)))
    (/
     (+ (- (/ 0.5 (* x x)) -1.0) (+ (/ 0.75 (* t_0 x)) (/ 1.875 (* t_0 t_0))))
     (* (sqrt x) (sqrt x))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / (sqrt(x) * sqrt(x)));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / (Math.sqrt(x) * Math.sqrt(x)));
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / (math.sqrt(x) * math.sqrt(x)))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(Float64(-x)) ^ Float64(-x))) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) + Float64(Float64(0.75 / Float64(t_0 * x)) + Float64(1.875 / Float64(t_0 * t_0)))) / Float64(sqrt(x) * sqrt(x))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((1.0 / sqrt(pi)) * (exp(-x) ^ -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / (sqrt(x) * sqrt(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[(-x)], $MachinePrecision], (-x)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] + N[(N[(0.75 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[x], $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-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs-rev N/A

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

   10. sqr-neg-rev N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp (- x)) (- x)))
    (/
     (+ (- (/ 0.5 (* x x)) -1.0) (+ (/ 0.75 (* t_0 x)) (/ 1.875 (* t_0 t_0))))
     (fabs x)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / fabs(x));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / Math.abs(x));
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / math.fabs(x))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(Float64(-x)) ^ Float64(-x))) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) + Float64(Float64(0.75 / Float64(t_0 * x)) + Float64(1.875 / Float64(t_0 * t_0)))) / abs(x)))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((1.0 / sqrt(pi)) * (exp(-x) ^ -x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / abs(x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs-rev N/A

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

   10. sqr-neg-rev N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp x) x))
    (/
     (+ (- (/ 0.5 (* x x)) -1.0) (+ (/ 0.75 (* t_0 x)) (/ 1.875 (* t_0 t_0))))
     (fabs x)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / fabs(x));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp(x), x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / Math.abs(x));
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(x), x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / math.fabs(x))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) + Float64(Float64(0.75 / Float64(t_0 * x)) + Float64(1.875 / Float64(t_0 * t_0)))) / abs(x)))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((1.0 / sqrt(pi)) * (exp(x) ^ x)) * ((((0.5 / (x * x)) - -1.0) + ((0.75 / (t_0 * x)) + (1.875 / (t_0 * t_0)))) / abs(x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 4: 99.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \frac{e^{x \cdot x}}{\frac{\left|x\right|}{\frac{0.5}{x \cdot x} - \left(-1 - \left(\frac{0.75}{t\_0} - \frac{-1.875}{\left(t\_0 \cdot x\right) \cdot x}\right)\right)} \cdot \sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (/
    (exp (* x x))
    (*
     (/
      (fabs x)
      (- (/ 0.5 (* x x)) (- -1.0 (- (/ 0.75 t_0) (/ -1.875 (* (* t_0 x) x))))))
     (sqrt PI)))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return exp((x * x)) / ((fabs(x) / ((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x)))))) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return Math.exp((x * x)) / ((Math.abs(x) / ((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x)))))) * Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = ((x * x) * x) * x
	return math.exp((x * x)) / ((math.fabs(x) / ((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x)))))) * math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	return Float64(exp(Float64(x * x)) / Float64(Float64(abs(x) / Float64(Float64(0.5 / Float64(x * x)) - Float64(-1.0 - Float64(Float64(0.75 / t_0) - Float64(-1.875 / Float64(Float64(t_0 * x) * x)))))) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * x) * x;
	tmp = exp((x * x)) / ((abs(x) / ((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x)))))) * sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Abs[x], $MachinePrecision] / N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(N[(0.75 / t$95$0), $MachinePrecision] - N[(-1.875 / N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \frac{\left(\frac{0.5}{x \cdot x} - \left(-1 - \left(\frac{0.75}{t\_0} - \frac{-1.875}{\left(t\_0 \cdot x\right) \cdot x}\right)\right)\right) \cdot e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (/
    (*
     (- (/ 0.5 (* x x)) (- -1.0 (- (/ 0.75 t_0) (/ -1.875 (* (* t_0 x) x)))))
     (exp (* x x)))
    (* (fabs x) (sqrt PI)))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return (((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x))))) * exp((x * x))) / (fabs(x) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return (((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x))))) * Math.exp((x * x))) / (Math.abs(x) * Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = ((x * x) * x) * x
	return (((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x))))) * math.exp((x * x))) / (math.fabs(x) * math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - Float64(-1.0 - Float64(Float64(0.75 / t_0) - Float64(-1.875 / Float64(Float64(t_0 * x) * x))))) * exp(Float64(x * x))) / Float64(abs(x) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * x) * x;
	tmp = (((0.5 / (x * x)) - (-1.0 - ((0.75 / t_0) - (-1.875 / ((t_0 * x) * x))))) * exp((x * x))) / (abs(x) * sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(N[(0.75 / t$95$0), $MachinePrecision] - N[(-1.875 / N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[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-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.9%

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

Alternative 6: 99.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (fma t_0 (/ 0.75 (pow x 4.0)) (* (+ (/ 0.5 (* x x)) 1.0) t_0))
    (/ (exp (* x x)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return fma(t_0, (0.75 / pow(x, 4.0)), (((0.5 / (x * x)) + 1.0) * t_0)) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(fma(t_0, Float64(0.75 / (x ^ 4.0)), Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) * t_0)) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (exp
  (*
   (log (/ (sqrt PI) (* (/ (- 1.0 (/ -0.5 (* x x))) (fabs x)) (exp (* x x)))))
   -1.0)))

C

double code(double x) {
	return exp((log((sqrt(((double) M_PI)) / (((1.0 - (-0.5 / (x * x))) / fabs(x)) * exp((x * x))))) * -1.0));
}

Java

public static double code(double x) {
	return Math.exp((Math.log((Math.sqrt(Math.PI) / (((1.0 - (-0.5 / (x * x))) / Math.abs(x)) * Math.exp((x * x))))) * -1.0));
}

Python

def code(x):
	return math.exp((math.log((math.sqrt(math.pi) / (((1.0 - (-0.5 / (x * x))) / math.fabs(x)) * math.exp((x * x))))) * -1.0))

Julia

function code(x)
	return exp(Float64(log(Float64(sqrt(pi) / Float64(Float64(Float64(1.0 - Float64(-0.5 / Float64(x * x))) / abs(x)) * exp(Float64(x * x))))) * -1.0))
end

MATLAB

function tmp = code(x)
	tmp = exp((log((sqrt(pi) / (((1.0 - (-0.5 / (x * x))) / abs(x)) * exp((x * x))))) * -1.0));
end

Wolfram

code[x_] := N[Exp[N[(N[Log[N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[(N[(1.0 - N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-fabs.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

10. Applied rewrites 99.6%

    \[\leadsto \color{blue}{e^{\log \left(\frac{\sqrt{\pi}}{\frac{1 - \frac{-0.5}{x \cdot x}}{\left|x\right|} \cdot e^{x \cdot x}}\right) \cdot -1}} \]
11. Add Preprocessing

Alternative 8: 99.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ (- 1.0 (/ -0.5 (* x x))) (fabs x)) (exp (* (- x) x))) (sqrt PI)))

C

double code(double x) {
	return (((1.0 - (-0.5 / (x * x))) / fabs(x)) / exp((-x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (((1.0 - (-0.5 / (x * x))) / Math.abs(x)) / Math.exp((-x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (((1.0 - (-0.5 / (x * x))) / math.fabs(x)) / math.exp((-x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 - Float64(-0.5 / Float64(x * x))) / abs(x)) / exp(Float64(Float64(-x) * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 - (-0.5 / (x * x))) / abs(x)) / exp((-x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 - N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-fabs.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. sqr-neg-rev N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. pow-neg N/A

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

10. Applied rewrites 99.6%

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

Alternative 9: 99.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)) (exp (* x x))) (sqrt PI)))

C

double code(double x) {
	return ((((0.5 / (x * x)) - -1.0) / fabs(x)) * exp((x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((((0.5 / (x * x)) - -1.0) / Math.abs(x)) * Math.exp((x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((((0.5 / (x * x)) - -1.0) / math.fabs(x)) * math.exp((x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) / abs(x)) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((((0.5 / (x * x)) - -1.0) / abs(x)) * exp((x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-fabs.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} \cdot e^{x \cdot x}}{\sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 10: 1.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot x\right)}^{-3.5} \cdot \frac{1.875}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow (* x x) -3.5) (/ 1.875 (sqrt PI))))

C

double code(double x) {
	return pow((x * x), -3.5) * (1.875 / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return Math.pow((x * x), -3.5) * (1.875 / Math.sqrt(Math.PI));
}

Python

def code(x):
	return math.pow((x * x), -3.5) * (1.875 / math.sqrt(math.pi))

Julia

function code(x)
	return Float64((Float64(x * x) ^ -3.5) * Float64(1.875 / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) ^ -3.5) * (1.875 / sqrt(pi));
end

Wolfram

code[x_] := N[(N[Power[N[(x * x), $MachinePrecision], -3.5], $MachinePrecision] * N[(1.875 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

   7. lower-fabs.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   9. lower-PI.f64 1.7

      \[\leadsto \frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

6. Applied rewrites 1.7%

   \[\leadsto \color{blue}{\frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\pi}\right)} \]

   3. metadata-eval N/A

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

   4. pow-prod-up N/A

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

   5. pow3 N/A

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

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

   7. pow3 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left|\color{blue}{x}\right| \cdot \sqrt{\pi}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\pi}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)} \]

   15. associate-*r* N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{x}\right)} \]

   16. associate-*r* N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{x}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{x}} \]

8. Applied rewrites 1.7%

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

10. Applied rewrites 1.7%

    \[\leadsto {\left(x \cdot x\right)}^{-3.5} \cdot \color{blue}{\frac{1.875}{\sqrt{\pi}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))))))