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

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 10

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, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   1.875
   (pow (fabs x) (- 7.0))
   (+
    (fma 0.75 (pow (fabs x) (- 5.0)) (/ 0.5 (fabs (* x (* x x)))))
    (/ 1.0 (fabs x))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(1.875, pow(fabs(x), -7.0), (fma(0.75, pow(fabs(x), -5.0), (0.5 / fabs((x * (x * x))))) + (1.0 / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(1.875, (abs(x) ^ Float64(-7.0)), Float64(fma(0.75, (abs(x) ^ Float64(-5.0)), Float64(0.5 / abs(Float64(x * Float64(x * x))))) + Float64(1.0 / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.875 * N[Power[N[Abs[x], $MachinePrecision], (-7.0)], $MachinePrecision] + N[(N[(0.75 * N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] + N[(0.5 / N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   1.875
   (pow (fabs x) (- 7.0))
   (fma
    (pow (fabs x) (- 5.0))
    0.75
    (+ (/ 0.5 (fabs (* x (* x x)))) (/ 1.0 (fabs x)))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma(1.875, pow(fabs(x), -7.0), fma(pow(fabs(x), -5.0), 0.75, ((0.5 / fabs((x * (x * x)))) + (1.0 / fabs(x)))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(1.875, (abs(x) ^ Float64(-7.0)), fma((abs(x) ^ Float64(-5.0)), 0.75, Float64(Float64(0.5 / abs(Float64(x * Float64(x * x)))) + Float64(1.0 / abs(x))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.875 * N[Power[N[Abs[x], $MachinePrecision], (-7.0)], $MachinePrecision] + N[(N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] * 0.75 + N[(N[(0.5 / N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $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. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (fma
   1.875
   (pow (fabs x) (- 7.0))
   (fma
    (pow (fabs x) (- 5.0))
    0.75
    (+ (/ 1.0 (fabs x)) (/ 0.5 (fabs (* x (* x x)))))))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(1.875, pow(fabs(x), -7.0), fma(pow(fabs(x), -5.0), 0.75, ((1.0 / fabs(x)) + (0.5 / fabs((x * (x * x)))))));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma(1.875, (abs(x) ^ Float64(-7.0)), fma((abs(x) ^ Float64(-5.0)), 0.75, Float64(Float64(1.0 / abs(x)) + Float64(0.5 / abs(Float64(x * Float64(x * x))))))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.875 * N[Power[N[Abs[x], $MachinePrecision], (-7.0)], $MachinePrecision] + N[(N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] * 0.75 + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \mathsf{fma}\left(0.75, {\left(\left|x\right|\right)}^{\left(-5\right)}, \mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left|x\right|}, 0.5, \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (pow (exp x) x))
  (fma
   0.75
   (pow (fabs x) (- 5.0))
   (fma (/ 1.0 (* (* x x) (fabs x))) 0.5 (/ 1.0 (fabs x))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * fma(0.75, pow(fabs(x), -5.0), fma((1.0 / ((x * x) * fabs(x))), 0.5, (1.0 / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * fma(0.75, (abs(x) ^ Float64(-5.0)), fma(Float64(1.0 / Float64(Float64(x * x) * abs(x))), 0.5, Float64(1.0 / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] + N[(N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\frac{1}{\pi}} \cdot e^{x \cdot x}\right) \cdot \mathsf{fma}\left(0.75, {\left(\left|x\right|\right)}^{\left(-5\right)}, \mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left|x\right|}, 0.5, \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (sqrt (/ 1.0 PI)) (exp (* x x)))
  (fma
   0.75
   (pow (fabs x) (- 5.0))
   (fma (/ 1.0 (* (* x x) (fabs x))) 0.5 (/ 1.0 (fabs x))))))

C

double code(double x) {
	return (sqrt((1.0 / ((double) M_PI))) * exp((x * x))) * fma(0.75, pow(fabs(x), -5.0), fma((1.0 / ((x * x) * fabs(x))), 0.5, (1.0 / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / pi)) * exp(Float64(x * x))) * fma(0.75, (abs(x) ^ Float64(-5.0)), fma(Float64(1.0 / Float64(Float64(x * x) * abs(x))), 0.5, Float64(1.0 / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] + N[(N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 1.8%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.7%

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

Alternative 6: 85.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \frac{\frac{0.5}{x \cdot x}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.75, {\left(\left|x\right|\right)}^{\left(-5\right)}, \mathsf{fma}\left(1.875, {\left(\left|x\right|\right)}^{\left(-7\right)}, \frac{1}{\left|x\right|}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.9e+107)
   (* (* (/ 1.0 (sqrt PI)) (pow (exp x) x)) (/ (/ 0.5 (* x x)) (fabs x)))
   (*
    (* (fma x x 1.0) (sqrt (/ 1.0 PI)))
    (fma
     0.75
     (pow (fabs x) (- 5.0))
     (fma 1.875 (pow (fabs x) (- 7.0)) (/ 1.0 (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.9e+107) {
		tmp = ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * ((0.5 / (x * x)) / fabs(x));
	} else {
		tmp = (fma(x, x, 1.0) * sqrt((1.0 / ((double) M_PI)))) * fma(0.75, pow(fabs(x), -5.0), fma(1.875, pow(fabs(x), -7.0), (1.0 / fabs(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.9e+107)
		tmp = Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * Float64(Float64(0.5 / Float64(x * x)) / abs(x)));
	else
		tmp = Float64(Float64(fma(x, x, 1.0) * sqrt(Float64(1.0 / pi))) * fma(0.75, (abs(x) ^ Float64(-5.0)), fma(1.875, (abs(x) ^ Float64(-7.0)), Float64(1.0 / abs(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.9e+107], N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x + 1.0), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[Abs[x], $MachinePrecision], (-5.0)], $MachinePrecision] + N[(1.875 * N[Power[N[Abs[x], $MachinePrecision], (-7.0)], $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9000000000000004e107

   1. Initial program 99.9%

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

   10. Applied rewrites 98.9%

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

   if 5.9000000000000004e107 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 78.1%

       \[\leadsto \left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(\color{blue}{0.75}, {\left(\left|x\right|\right)}^{\left(-5\right)}, \mathsf{fma}\left(1.875, {\left(\left|x\right|\right)}^{\left(-7\right)}, \frac{1}{\left|x\right|}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 33.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.4%

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

10. Applied rewrites 33.9%

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

Alternative 8: 32.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.4%

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

10. Applied rewrites 32.4%

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

Alternative 9: 1.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 1.8%

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

10. Applied rewrites 1.8%

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{\pi}}}{x \cdot x}}{\left|x\right|} \cdot 0.5 \]
11. Add Preprocessing

Alternative 10: 1.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 1.8%

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

10. Applied rewrites 1.8%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left(x \cdot x\right) \cdot \left|x\right|} \cdot 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025121 
(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)))))))