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

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 13

Speedup: 2.7×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

12. Applied rewrites 100.0%

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

13. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 100.0

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

15. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

    1. lift-pow.f64 N/A

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

    2. lift-exp.f64 N/A

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

    3. pow-exp N/A

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

    4. lift-fabs.f64 N/A

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

    5. lift-neg.f64 N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-neg.f64 N/A

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

    8. sqr-neg-rev N/A

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

    9. exp-prod N/A

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

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

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

    11. pow2 N/A

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

    12. sqrt-pow1 N/A

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

    13. metadata-eval N/A

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

    14. unpow1 N/A

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

    15. lower-pow.f64 N/A

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

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

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

    17. pow2 N/A

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

    18. sqrt-pow1 N/A

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

    19. metadata-eval N/A

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

    20. unpow1 N/A

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

    21. lower-exp.f64 100.0

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

11. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval 99.6

       \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

10. Taylor expanded in x around -inf

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

12. Applied rewrites 100.0%

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

Alternative 4: 99.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 99.7

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 6: 99.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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%

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

7. Taylor expanded in x around -inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval 99.6

       \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

11. Applied rewrites 99.6%

    \[\leadsto \frac{\frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot e^{x \cdot x}}{\color{blue}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 8: 99.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval 99.6

       \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 9: 99.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (exp (+ 0.0 (* x x))) (sqrt PI)) x))

C

double code(double x) {
	return (exp((0.0 + (x * x))) / sqrt(((double) M_PI))) / x;
}

Java

public static double code(double x) {
	return (Math.exp((0.0 + (x * x))) / Math.sqrt(Math.PI)) / x;
}

Python

def code(x):
	return (math.exp((0.0 + (x * x))) / math.sqrt(math.pi)) / x

Julia

function code(x)
	return Float64(Float64(exp(Float64(0.0 + Float64(x * x))) / sqrt(pi)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (exp((0.0 + (x * x))) / sqrt(pi)) / x;
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(0.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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%

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

7. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{\color{blue}{x}} \]
10. Add Preprocessing

Alternative 10: 99.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0 + x \cdot x}}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp (+ 0.0 (* x x))) (* x (sqrt PI))))

C

double code(double x) {
	return exp((0.0 + (x * x))) / (x * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return Math.exp((0.0 + (x * x))) / (x * Math.sqrt(Math.PI));
}

Python

def code(x):
	return math.exp((0.0 + (x * x))) / (x * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(exp(Float64(0.0 + Float64(x * x))) / Float64(x * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = exp((0.0 + (x * x))) / (x * sqrt(pi));
end

Wolfram

code[x_] := N[(N[Exp[N[(0.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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. Taylor expanded in x around inf

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

12. Applied rewrites 99.5%

    \[\leadsto \frac{e^{0 + x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \]
13. Add Preprocessing

Alternative 11: 2.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval 99.6

       \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 2.3%

    \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\color{blue}{\pi}}} \]
13. Add Preprocessing

Alternative 12: 1.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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-PI.f64 N/A

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

    8. lift-sqrt.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. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. associate-*r/ N/A

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

    14. lower-/.f64 N/A

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

11. Applied rewrites 1.8%

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

Alternative 13: 1.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

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

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

7. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))