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

Percentage Accurate: 100.0% → 100.0%

Time: 13.8s

Alternatives: 14

Speedup: 1.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2024226 
(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)))))))