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

Percentage Accurate: 100.0% → 100.0%

Time: 20.1s

Alternatives: 4

Speedup: 1.6×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\frac{\frac{\frac{0.5}{x}}{x} + 1}{x} + \frac{3}{4} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{x}\right), x\right) \cdot t\_0\right)\right) + \frac{15}{8} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{x \cdot x}\right), x\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* x x)))
    (+
     (+
      (/ (+ (/ (/ 0.5 x) x) 1.0) x)
      (* (/ 3.0 4.0) (* (ratio-of-squares (/ 1.0 x) x) t_0)))
     (* (/ 15.0 8.0) (* (ratio-of-squares (/ -1.0 (* x x)) x) t_0))))))

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. associate-*r/ N/A

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

   8. frac-times N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. associate-*l* N/A

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

   13. frac-times N/A

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

   14. metadata-eval N/A

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

   15. sqr-abs-rev N/A

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

   16. pow2 N/A

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

   17. frac-times N/A

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-fabs.f64 N/A

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

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

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-fabs.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-fabs.f64 N/A

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

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\frac{\frac{0.5}{x}}{x} + 1}{x}} + \frac{3}{4} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-ratio-of-squares.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. associate-*l* N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. pow2 N/A

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

   13. pow-flip N/A

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

   14. pow2 N/A

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

   15. pow-div N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. pow-flip N/A

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

   20. frac-times N/A

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

   21. metadata-eval N/A

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

   22. sqr-abs-rev N/A

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

   23. pow2 N/A

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

10. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{\frac{0.5}{x}}{x} + 1}{x} + \frac{3}{4} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\color{blue}{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{x \cdot x}\right), x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\frac{\frac{\frac{0.5}{x}}{x} + 1}{x} + \frac{3}{4} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{x \cdot x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. associate-*r/ N/A

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

   8. frac-times N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. associate-*l* N/A

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

   13. frac-times N/A

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

   14. metadata-eval N/A

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

   15. sqr-abs-rev N/A

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

   16. pow2 N/A

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

   17. frac-times N/A

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-fabs.f64 N/A

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

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

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-fabs.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-fabs.f64 N/A

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

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\frac{\frac{0.5}{x}}{x} + 1}{x}} + \frac{3}{4} \cdot \left(\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{x}\right), x\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 34.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * (((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[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\left(\frac{0.5}{x \cdot x} + 1\right) \cdot {\left(\left|x\right|\right)}^{-1}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. frac-times N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. frac-times N/A

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

   9. associate-*r* N/A

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

7. Applied rewrites 32.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. associate-*r/ N/A

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

   7. sqr-pow N/A

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

   8. pow-prod-down N/A

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

   9. sqr-abs-rev N/A

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

   10. pow-prod-down N/A

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

   11. sqr-pow N/A

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

   12. metadata-eval N/A

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

   13. unpow3 N/A

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

   14. pow2 N/A

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

   15. associate-/r* N/A

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

   16. pow2 N/A

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

   17. sqr-abs-rev N/A

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lower-/.f64 N/A

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

   24. associate-/r* N/A

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

   25. lower-/.f64 N/A

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

   26. lower-/.f64 32.5

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

9. Applied rewrites 32.5%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-exp.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fabs.f64 N/A

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

    6. lift-fabs.f64 N/A

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

    7. associate-*l/ N/A

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

    8. lower-/.f64 N/A

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

    9. 1-exp N/A

       \[\leadsto \frac{\color{blue}{e^{0}} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    10. sqr-abs-rev N/A

        \[\leadsto \frac{e^{0} \cdot e^{\color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    11. pow2 N/A

        \[\leadsto \frac{e^{0} \cdot e^{\color{blue}{{x}^{2}}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    12. prod-exp N/A

        \[\leadsto \frac{\color{blue}{e^{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    13. lower-exp.f64 N/A

        \[\leadsto \frac{\color{blue}{e^{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    14. lower-+.f64 N/A

        \[\leadsto \frac{e^{\color{blue}{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    15. pow2 N/A

        \[\leadsto \frac{e^{0 + \color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{x} \]

    16. lower-*.f64 32.5

        \[\leadsto \frac{e^{0 + \color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{0.5}{x}}{x}}{x} \]

11. Applied rewrites 32.5%

    \[\leadsto \color{blue}{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}} \cdot \frac{\frac{\frac{0.5}{x}}{x}}{x} \]

12. Final simplification 32.5%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\frac{\frac{0.5}{x}}{x}}{x} \]
13. Add Preprocessing

Alternative 4: 32.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / N[(N[((-x) * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * 0.5), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\left(\frac{0.5}{x \cdot x} + 1\right) \cdot {\left(\left|x\right|\right)}^{-1}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. frac-times N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. frac-times N/A

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

   9. associate-*r* N/A

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

7. Applied rewrites 32.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. unpow3 N/A

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

   6. sqr-abs-rev N/A

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

   7. pow2 N/A

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

   8. frac-times N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. pow2 N/A

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

   12. sqr-abs-rev N/A

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

   13. distribute-lft-neg-out N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. frac-times N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. lower-*.f64 30.9

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

9. Applied rewrites 30.9%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-exp.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fabs.f64 N/A

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

    6. lift-fabs.f64 N/A

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

    7. associate-*l/ N/A

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

    8. lower-/.f64 N/A

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

    9. 1-exp N/A

       \[\leadsto \frac{\color{blue}{e^{0}} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    10. sqr-abs-rev N/A

        \[\leadsto \frac{e^{0} \cdot e^{\color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    11. pow2 N/A

        \[\leadsto \frac{e^{0} \cdot e^{\color{blue}{{x}^{2}}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    12. prod-exp N/A

        \[\leadsto \frac{\color{blue}{e^{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    13. lower-exp.f64 N/A

        \[\leadsto \frac{\color{blue}{e^{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    14. lower-+.f64 N/A

        \[\leadsto \frac{e^{\color{blue}{0 + {x}^{2}}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    15. pow2 N/A

        \[\leadsto \frac{e^{0 + \color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot \frac{1}{2}\right) \]

    16. lower-*.f64 30.9

        \[\leadsto \frac{e^{0 + \color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot 0.5\right) \]

11. Applied rewrites 30.9%

    \[\leadsto \color{blue}{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot x} \cdot 0.5\right) \]

12. Final simplification 30.9%

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

Reproduce

herbie shell --seed 2025065 
(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)))))))