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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 12

Speedup: 2.4×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ t_1 := -\left|x\right|\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{t\_1}\right)}^{t\_1}\right) \cdot \left(\left(\frac{0.5}{t\_0} + \left(\frac{1}{x} - \frac{-0.75}{\left(t\_0 \cdot x\right) \cdot x}\right)\right) + {x}^{-7} \cdot 1.875\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)) (t_1 (- (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp t_1) t_1))
    (+
     (+ (/ 0.5 t_0) (- (/ 1.0 x) (/ -0.75 (* (* t_0 x) x))))
     (* (pow x -7.0) 1.875)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = -fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(t_1), t_1)) * (((0.5 / t_0) + ((1.0 / x) - (-0.75 / ((t_0 * x) * x)))) + (pow(x, -7.0) * 1.875));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = -Math.abs(x);
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp(t_1), t_1)) * (((0.5 / t_0) + ((1.0 / x) - (-0.75 / ((t_0 * x) * x)))) + (Math.pow(x, -7.0) * 1.875));
}

Python

def code(x):
	t_0 = (x * x) * x
	t_1 = -math.fabs(x)
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(t_1), t_1)) * (((0.5 / t_0) + ((1.0 / x) - (-0.75 / ((t_0 * x) * x)))) + (math.pow(x, -7.0) * 1.875))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	t_1 = Float64(-abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(t_1) ^ t_1)) * Float64(Float64(Float64(0.5 / t_0) + Float64(Float64(1.0 / x) - Float64(-0.75 / Float64(Float64(t_0 * x) * x)))) + Float64((x ^ -7.0) * 1.875)))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	t_1 = -abs(x);
	tmp = ((1.0 / sqrt(pi)) * (exp(t_1) ^ t_1)) * (((0.5 / t_0) + ((1.0 / x) - (-0.75 / ((t_0 * x) * x)))) + ((x ^ -7.0) * 1.875));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[x], $MachinePrecision])}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[t$95$1], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(-0.75 / N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -7.0], $MachinePrecision] * 1.875), $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-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

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%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{-\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{-3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right)\right) + \color{blue}{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}}\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{-\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{-3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right)\right) + {\left(\left|x\right|\right)}^{-7} \cdot \frac{15}{8}\right) \]

   6. sqr-pow N/A

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

   7. pow-prod-down N/A

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

   8. sqr-abs-rev N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{-\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{-3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right)\right) + {\left(x \cdot x\right)}^{\left(\frac{-7}{2}\right)} \cdot \frac{15}{8}\right) \]

   9. pow-prod-down N/A

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

   10. sqr-pow N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left|x\right|\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{t\_0}\right)}^{t\_0}\right) \cdot \mathsf{fma}\left({x}^{-7}, 1.875, -\frac{\left(-\frac{\frac{0.75}{x \cdot x} + 0.5}{x \cdot x}\right) - 1}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp t_0) t_0))
    (fma
     (pow x -7.0)
     1.875
     (- (/ (- (- (/ (+ (/ 0.75 (* x x)) 0.5) (* x x))) 1.0) x))))))

C

double code(double x) {
	double t_0 = -fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(t_0), t_0)) * fma(pow(x, -7.0), 1.875, -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x));
}

Julia

function code(x)
	t_0 = Float64(-abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(t_0) ^ t_0)) * fma((x ^ -7.0), 1.875, Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.75 / Float64(x * x)) + 0.5) / Float64(x * x))) - 1.0) / x))))
end

Wolfram

code[x_] := Block[{t$95$0 = (-N[Abs[x], $MachinePrecision])}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[t$95$0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -7.0], $MachinePrecision] * 1.875 + (-N[(N[((-N[(N[(N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]) - 1.0), $MachinePrecision] / 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-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

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%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in x around -inf

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

11. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((1.0 / sqrt(pi)) * exp((x * x))) * ((1.0 / x) - ((-0.5 / t_0) - (((1.875 / (x * x)) + 0.75) / (t_0 * (x * 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[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(N[(-0.5 / t$95$0), $MachinePrecision] - N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{\color{blue}{{x}^{5}}}\right)\right) \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{5}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{5}}\right)\right) \]

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8} \cdot 1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8} \cdot 1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   12. pow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   15. metadata-eval N/A

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

   16. pow-prod-up N/A

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

   17. pow3 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

12. Applied rewrites 100.0%

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

13. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

15. Applied rewrites 100.0%

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

16. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

18. Applied rewrites 100.0%

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

19. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

21. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{\color{blue}{{x}^{5}}}\right)\right) \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{5}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{5}}\right)\right) \]

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8} \cdot 1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8} \cdot 1}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{{x}^{2}} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   12. pow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \left(\frac{\frac{-1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{5}}\right)\right) \]

   15. metadata-eval N/A

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

   16. pow-prod-up N/A

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

   17. pow3 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

Alternative 6: 99.7% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (exp (* x x))
   (- (/ 1.0 x) (/ (fma -0.5 x (/ -0.75 x)) (* (* (* x x) x) x))))
  (sqrt PI)))

C

double code(double x) {
	return (exp((x * x)) * ((1.0 / x) - (fma(-0.5, x, (-0.75 / x)) / (((x * x) * x) * x)))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) * Float64(Float64(1.0 / x) - Float64(fma(-0.5, x, Float64(-0.75 / x)) / Float64(Float64(Float64(x * x) * x) * x)))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(N[(-0.5 * x + N[(-0.75 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $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%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

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

10. Applied rewrites 99.7%

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

Alternative 7: 99.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((1.0 / x) - -(((0.75 / (x * x)) + 0.5) / ((x * x) * x)));
}

Java

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

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * ((1.0 / x) - -(((0.75 / (x * x)) + 0.5) / ((x * x) * x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. mul-1-neg N/A

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

    3. lower-neg.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. +-commutative N/A

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

    6. lower-+.f64 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. metadata-eval N/A

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

    11. lower-/.f64 N/A

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

    12. metadata-eval N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 N/A

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

11. Applied rewrites 99.7%

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

Alternative 8: 99.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{x} - \frac{\frac{-1}{2}}{\color{blue}{{x}^{3}}}\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. pow3 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 9: 99.6% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. sqr-abs-rev N/A

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

   3. pow2 N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 10: 99.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(x * 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%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. sqr-abs-rev N/A

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

   3. pow2 N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-exp.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-/l/ N/A

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

    6. sqr-abs-rev N/A

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

    7. pow2 N/A

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

    8. *-commutative N/A

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

    9. lift-PI.f64 N/A

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

    10. lift-sqrt.f64 N/A

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

11. Applied rewrites 99.5%

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

Alternative 11: 52.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. sqr-abs-rev N/A

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

   3. pow2 N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{1 + {x}^{2}}{\sqrt{\pi}}}{x} \]
11. Step-by-step derivation

    1. sqr-abs-rev N/A

       \[\leadsto \frac{\frac{1 + {x}^{2}}{\sqrt{\pi}}}{x} \]

    2. +-commutative N/A

       \[\leadsto \frac{\frac{{x}^{2} + 1}{\sqrt{\pi}}}{x} \]

    3. pow2 N/A

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

    4. lower-fma.f64 52.8

       \[\leadsto \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x} \]

12. Applied rewrites 52.8%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{x} \]
13. Add Preprocessing

Alternative 12: 2.3% accurate, 17.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. sqr-abs-rev N/A

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

   3. pow2 N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in x around 0

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

    1. sqr-abs-rev 2.3

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

12. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2025120 
(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)))))))