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

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 10

Speedup: 2.2×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ t_1 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(1 - \frac{\frac{-0.5}{t\_0}}{t\_1}, t\_1, {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right) + \frac{0.75}{\left(t\_0 \cdot x\right) \cdot \left|x\right|}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)) (t_1 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (* (exp x) (exp x)) (/ x 2.0)))
    (+
     (fma (- 1.0 (/ (/ -0.5 t_0) t_1)) t_1 (* (pow (fabs x) -7.0) 1.875))
     (/ 0.75 (* (* t_0 x) (fabs x)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = 1.0 / fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * pow((exp(x) * exp(x)), (x / 2.0))) * (fma((1.0 - ((-0.5 / t_0) / t_1)), t_1, (pow(fabs(x), -7.0) * 1.875)) + (0.75 / ((t_0 * x) * fabs(x))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	t_1 = Float64(1.0 / abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (Float64(exp(x) * exp(x)) ^ Float64(x / 2.0))) * Float64(fma(Float64(1.0 - Float64(Float64(-0.5 / t_0) / t_1)), t_1, Float64((abs(x) ^ -7.0) * 1.875)) + Float64(0.75 / Float64(Float64(t_0 * x) * abs(x)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Exp[x], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[(N[(-0.5 / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(N[(t$95$0 * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

8. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x} \cdot e^{x}\right)}^{\color{blue}{\left(\frac{x}{2}\right)}}\right) \cdot \left(\mathsf{fma}\left(1 - \frac{\frac{-0.5}{\left(x \cdot x\right) \cdot x}}{\frac{1}{\left|x\right|}}, \frac{1}{\left|x\right|}, {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right) + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) \]

10. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}}\right) \cdot \left(\mathsf{fma}\left(1 - \frac{\frac{-0.5}{\left(x \cdot x\right) \cdot x}}{\frac{1}{\left|x\right|}}, \frac{1}{\left|x\right|}, {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right) + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. div-sub N/A

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

    2. pow-div N/A

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

    3. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\mathsf{fma}\left({x}^{0} - \frac{\frac{-1}{2} \cdot \left|x\right|}{{x}^{3}}, \frac{1}{\left|x\right|}, {\left(\left|x\right|\right)}^{-7} \cdot \frac{15}{8}\right) + \frac{\frac{3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) \]

    4. metadata-eval N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. fp-cancel-sign-sub-inv N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

    11. lower-fma.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. lift-fabs.f64 N/A

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

    14. pow3 N/A

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

    15. lift-*.f64 N/A

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

    16. lift-*.f64 N/A

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

    17. metadata-eval 100.0

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

11. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{\color{blue}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied rewrites 100.0%

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

Alternative 6: 99.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 49.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x} - \frac{0.25}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}, \frac{1}{\frac{1}{\left|x\right|} \cdot \left(1 - \frac{0.5}{x \cdot x}\right)}, \frac{1}{\left|x\right|} \cdot \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right)\right)} \]

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 24.8%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 8: 99.6% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 49.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x} - \frac{0.25}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}, \frac{1}{\frac{1}{\left|x\right|} \cdot \left(1 - \frac{0.5}{x \cdot x}\right)}, \frac{1}{\left|x\right|} \cdot \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right)\right)} \]

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 24.8%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 9: 1.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. unpow3 N/A

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

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. pow-prod-down N/A

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

   12. sqr-abs-rev N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. unpow3 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lift-sqrt.f64 1.8

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 1.8%

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

    1. metadata-eval N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lift-PI.f64 N/A

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

    9. associate-/r* N/A

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

    10. pow2 N/A

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

    11. lower-/.f64 N/A

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

    12. pow2 N/A

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

    13. lower-/.f64 N/A

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

    14. metadata-eval N/A

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

    15. pow2 N/A

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

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lower-*.f64 N/A

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

    19. lift-PI.f64 N/A

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

    20. lift-sqrt.f64 1.8

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

11. Applied rewrites 1.8%

    \[\leadsto \frac{\frac{0.5}{x \cdot x}}{\sqrt{\pi} \cdot \color{blue}{x}} \]
12. Add Preprocessing

Alternative 10: 1.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. unpow3 N/A

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

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. pow-prod-down N/A

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

   12. sqr-abs-rev N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. unpow3 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lift-sqrt.f64 1.8

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 1.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-PI.f64 N/A

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

    12. lift-sqrt.f64 1.8

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

11. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025136 
(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)))))))