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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 9

Speedup: 2.3×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left|x\right|\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{t\_0}\right)}^{t\_0}\right) \cdot \mathsf{fma}\left(\frac{{x}^{-6}}{\left|x\right|}, 1.875, \frac{\frac{0.75}{\left|x\right| \cdot x} + 0.5}{\left(x \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp t_0) t_0))
    (fma
     (/ (pow x -6.0) (fabs x))
     1.875
     (+ (/ (+ (/ 0.75 (* (fabs x) x)) 0.5) (* (* x x) x)) (/ 1.0 (fabs x)))))))

C

double code(double x) {
	double t_0 = -fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(t_0), t_0)) * fma((pow(x, -6.0) / fabs(x)), 1.875, ((((0.75 / (fabs(x) * x)) + 0.5) / ((x * x) * x)) + (1.0 / fabs(x))));
}

Julia

function code(x)
	t_0 = Float64(-abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(t_0) ^ t_0)) * fma(Float64((x ^ -6.0) / abs(x)), 1.875, Float64(Float64(Float64(Float64(0.75 / Float64(abs(x) * x)) + 0.5) / Float64(Float64(x * x) * x)) + Float64(1.0 / abs(x)))))
end

Wolfram

code[x_] := Block[{t$95$0 = (-N[Abs[x], $MachinePrecision])}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[t$95$0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[x, -6.0], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.875 + N[(N[(N[(N[(0.75 / N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. pow3 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around -inf

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

   1. lower-+.f64 N/A

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

8. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

Alternative 5: 99.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. pow3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 6: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. sqr-abs-rev N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

6. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\mathsf{fma}\left(x, x, -4 \cdot \log x\right)}}{\left|x\right|}}{\sqrt{\pi}} \cdot 0.75 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (/ (exp (fma x x (* -4.0 (log x)))) (fabs x)) (sqrt PI)) 0.75))

C

double code(double x) {
	return ((exp(fma(x, x, (-4.0 * log(x)))) / fabs(x)) / sqrt(((double) M_PI))) * 0.75;
}

Julia

function code(x)
	return Float64(Float64(Float64(exp(fma(x, x, Float64(-4.0 * log(x)))) / abs(x)) / sqrt(pi)) * 0.75)
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[(x * x + N[(-4.0 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.75), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. associate-*r/ N/A

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

   3. pow2 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\left|x\right| \cdot \left|x\right|}}{{x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   4. sqr-abs-rev N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{x \cdot x}}{{x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   5. pow2 N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x - \log x \cdot 4}}{\left|x\right| \cdot \sqrt{\pi}} \cdot 0.75} \]

8. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{e^{\mathsf{fma}\left(x, x, -4 \cdot \log x\right)}}{\left|x\right|}}{\sqrt{\pi}} \cdot 0.75 \]
9. Add Preprocessing

Alternative 8: 99.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.6%

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

Alternative 9: 1.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. unpow3 N/A

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

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. pow-prod-down N/A

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

   12. sqr-abs-rev N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. pow3 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lift-sqrt.f64 1.8

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

7. Applied rewrites 1.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. pow3 N/A

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

   7. unpow3 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-sqrt.f64 1.8

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

9. Applied rewrites 1.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-PI.f64 N/A

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

    12. lift-sqrt.f64 1.8

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

11. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025138 
(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)))))))