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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 10

Speedup: 2.5×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-fabs.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. sqr-neg-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. sqr-neg-rev N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

8. Applied rewrites 100.0%

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

Alternative 4: 99.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

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

   1. lift-/.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|}\right)} \]
9. 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. metadata-eval N/A

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

   2. associate-/l* N/A

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

   3. pow2 N/A

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

   4. sqr-abs-rev N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 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: 99.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.exp(((x * x) - (Math.log(x) * 4.0))) / (Math.sqrt(Math.PI) * Math.abs(x))) * 0.75;
}

Python

def code(x):
	return (math.exp(((x * x) - (math.log(x) * 4.0))) / (math.sqrt(math.pi) * math.fabs(x))) * 0.75

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp(((x * x) - (log(x) * 4.0))) / (sqrt(pi) * abs(x))) * 0.75;
end

Wolfram

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

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

4. Taylor expanded in x around inf

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

   1. lift-/.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

   1. sqr-abs-rev 2.3

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

   2. sqr-neg-rev 2.3

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

   3. lift-neg.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. pow-exp N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-neg.f64 2.3

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

9. Applied rewrites 2.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{3}{4} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]
11. 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

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

12. Applied rewrites 99.5%

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

Alternative 8: 33.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. associate-*l* N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval 33.2

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. associate-*l* N/A

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

   14. sqr-abs-rev N/A

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

   15. cube-mult N/A

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

   16. metadata-eval N/A

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

   17. pow-prod-up N/A

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

   18. pow-prod-down N/A

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

   19. sqr-abs-rev N/A

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

   20. pow-prod-down N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{\frac{3}{2}} \cdot {x}^{\frac{3}{2}}} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   21. pow-prod-up N/A

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

   22. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{3}} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   23. pow3 N/A

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

   24. lift-*.f64 N/A

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

7. Applied rewrites 33.2%

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

Alternative 9: 1.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

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

9. Applied rewrites 1.8%

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

    1. metadata-eval N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. associate-/r* N/A

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

    10. mult-flip-rev N/A

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

    11. lower-/.f64 N/A

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

    12. mult-flip-rev N/A

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

    13. pow2 N/A

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

    14. lift-/.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. metadata-eval N/A

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

    17. *-commutative N/A

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

    18. lower-*.f64 N/A

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

    19. lift-sqrt.f64 N/A

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

    20. lift-PI.f64 1.8

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

11. Applied rewrites 1.8%

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

Alternative 10: 1.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

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

9. Applied rewrites 1.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-sqrt.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-sqrt.f64 N/A

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

    12. lift-PI.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 2025142 
(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)))))))