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

Percentage Accurate: 100.0% → 99.6%

Time: 11.4s

Alternatives: 12

Speedup: N/A×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 3.5× speedup

Math

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

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)) / abs(Float64(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[Abs[N[(x * N[Sqrt[Pi], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 99.6%

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

Alternative 3: 86.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{t\_0}{\left|x\right|} \cdot \frac{\mathsf{fma}\left(t\_2, t\_2, -1\right)}{\mathsf{fma}\left(x, t\_1, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI)))
        (t_1 (* x (fma 0.5 (* x x) 1.0)))
        (t_2 (* x t_1)))
   (if (<= (fabs x) 5e+58)
     (* (/ t_0 (fabs x)) (/ (fma t_2 t_2 -1.0) (fma x t_1 -1.0)))
     (* t_0 (* (fabs x) (* x (* x (* (* x x) 0.16666666666666666))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = x * fma(0.5, (x * x), 1.0);
	double t_2 = x * t_1;
	double tmp;
	if (fabs(x) <= 5e+58) {
		tmp = (t_0 / fabs(x)) * (fma(t_2, t_2, -1.0) / fma(x, t_1, -1.0));
	} else {
		tmp = t_0 * (fabs(x) * (x * (x * ((x * x) * 0.16666666666666666))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(x * fma(0.5, Float64(x * x), 1.0))
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (abs(x) <= 5e+58)
		tmp = Float64(Float64(t_0 / abs(x)) * Float64(fma(t_2, t_2, -1.0) / fma(x, t_1, -1.0)));
	else
		tmp = Float64(t_0 * Float64(abs(x) * Float64(x * Float64(x * Float64(Float64(x * x) * 0.16666666666666666)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e+58], N[(N[(t$95$0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision] / N[(x * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.99999999999999986e58

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-fabs.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 4.0%

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

   12. Applied rewrites 37.5%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right), -1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right), -1\right)} \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \]

   if 4.99999999999999986e58 < (fabs.f64 x)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-fabs.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}, \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right)}, \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{6} \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right), -1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (sqrt (/ 1.0 PI)) (fabs x))
  (fma x (* x (fma (* x x) (fma x (* x 0.16666666666666666) 0.5) 1.0)) 1.0)))

C

double code(double x) {
	return (sqrt((1.0 / ((double) M_PI))) / fabs(x)) * fma(x, (x * fma((x * x), fma(x, (x * 0.16666666666666666), 0.5), 1.0)), 1.0);
}

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / pi)) / abs(x)) * fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.16666666666666666), 0.5), 1.0)), 1.0))
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 50.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 82.5%

    \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{\left|x\right|} \]

14. Final simplification 82.5%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \]
15. Add Preprocessing

Alternative 5: 84.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|x \cdot \sqrt{\pi}\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma x (* x (fma (* x x) (fma x (* x 0.16666666666666666) 0.5) 1.0)) 1.0)
  (fabs (* x (sqrt PI)))))

C

double code(double x) {
	return fma(x, (x * fma((x * x), fma(x, (x * 0.16666666666666666), 0.5), 1.0)), 1.0) / fabs((x * sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.16666666666666666), 0.5), 1.0)), 1.0) / abs(Float64(x * sqrt(pi))))
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[N[(x * N[Sqrt[Pi], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 82.1%

    \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|\color{blue}{x \cdot \sqrt{\pi}}\right|} \]
13. Add Preprocessing

Alternative 6: 80.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma x (* x 0.16666666666666666) 0.5)
  (* (fabs x) (* (* x x) (sqrt (/ 1.0 PI))))))

C

double code(double x) {
	return fma(x, (x * 0.16666666666666666), 0.5) * (fabs(x) * ((x * x) * sqrt((1.0 / ((double) M_PI)))));
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * 0.16666666666666666), 0.5) * Float64(abs(x) * Float64(Float64(x * x) * sqrt(Float64(1.0 / pi)))))
end

Wolfram

code[x_] := N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 80.0%

    \[\leadsto \mathsf{fma}\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}, \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right)}, \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto {x}^{4} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{\left|x\right|}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right) \]
12. Step-by-step derivation

13. Applied rewrites 80.0%

    \[\leadsto \left(\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.16666666666666666}, 0.5\right) \]

14. Final simplification 80.0%

    \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]
15. Add Preprocessing

Alternative 7: 80.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (* (fabs x) (* x (* x (* (* x x) 0.16666666666666666))))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (fabs(x) * (x * (x * ((x * x) * 0.16666666666666666))));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (Math.abs(x) * (x * (x * ((x * x) * 0.16666666666666666))));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (math.fabs(x) * (x * (x * ((x * x) * 0.16666666666666666))))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * Float64(x * Float64(x * Float64(Float64(x * x) * 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (abs(x) * (x * (x * ((x * x) * 0.16666666666666666))));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 80.0%

    \[\leadsto \mathsf{fma}\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}, \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right)}, \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \frac{1}{6} \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 80.0%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}\right) \]
14. Add Preprocessing

Alternative 8: 75.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 74.9%

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

12. Applied rewrites 74.6%

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

Alternative 9: 51.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{\left|x\right|} \end{array} \]

FPCore

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

C

double code(double x) {
	return (fma(x, x, 1.0) / sqrt(((double) M_PI))) / fabs(x);
}

Julia

function code(x)
	return Float64(Float64(fma(x, x, 1.0) / sqrt(pi)) / abs(x))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 50.4%

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

12. Applied rewrites 50.4%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi}}}{\color{blue}{\left|x\right|}} \]
13. Add Preprocessing

Alternative 10: 51.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x + 1.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 50.4%

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

12. Applied rewrites 50.0%

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

Alternative 11: 2.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 2.3%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{\left|x\right|}} \]
11. Add Preprocessing

Alternative 12: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 2.3%

    \[\leadsto \frac{1}{\left|\color{blue}{x \cdot \sqrt{\pi}}\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024231 
(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)))))))