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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 10

Speedup: 2.8×

• 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-0.5}{x \cdot x}\\ t_1 := \frac{\frac{1.875}{x \cdot x} - -0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(-\frac{\frac{{t\_1}^{3} - {t\_0}^{3}}{{t\_1}^{2} + \mathsf{fma}\left(t\_0, t\_0, t\_1 \cdot t\_0\right)}}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -0.5 (* x x))))
        (t_1 (/ (- (/ 1.875 (* x x)) -0.75) (- (* (* (* x x) x) x)))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp x) x))
    (-
     (/
      (/
       (- (pow t_1 3.0) (pow t_0 3.0))
       (+ (pow t_1 2.0) (fma t_0 t_0 (* t_1 t_0))))
      x)))))

C

double code(double x) {
	double t_0 = 1.0 - (-0.5 / (x * x));
	double t_1 = ((1.875 / (x * x)) - -0.75) / -(((x * x) * x) * x);
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * -(((pow(t_1, 3.0) - pow(t_0, 3.0)) / (pow(t_1, 2.0) + fma(t_0, t_0, (t_1 * t_0)))) / x);
}

Julia

function code(x)
	t_0 = Float64(1.0 - Float64(-0.5 / Float64(x * x)))
	t_1 = Float64(Float64(Float64(1.875 / Float64(x * x)) - -0.75) / Float64(-Float64(Float64(Float64(x * x) * x) * x)))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * Float64(-Float64(Float64(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)) / Float64((t_1 ^ 2.0) + fma(t_0, t_0, Float64(t_1 * t_0)))) / x)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.75), $MachinePrecision] / (-N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision])), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * (-N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(-\frac{\frac{{\left(\frac{\frac{1.875}{x \cdot x} - -0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)}^{3} - {\left(1 - \frac{-0.5}{x \cdot x}\right)}^{3}}{{\left(\frac{\frac{1.875}{x \cdot x} - -0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)}^{2} + \mathsf{fma}\left(1 - \frac{-0.5}{x \cdot x}, 1 - \frac{-0.5}{x \cdot x}, \frac{\frac{1.875}{x \cdot x} - -0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x} \cdot \left(1 - \frac{-0.5}{x \cdot x}\right)\right)}}{x}\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (exp (* (log (sqrt PI)) -1.0)) (pow (exp x) x))
  (-
   (/
    (-
     (- (/ (+ (/ 1.875 (* x x)) 0.75) (* (* (* x x) x) x)))
     (- 1.0 (/ -0.5 (* x x))))
    x))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

    1. lift-/.f64 N/A

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

    2. inv-pow N/A

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

    3. pow-to-exp N/A

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

    4. lower-exp.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-log.f64 100.0

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

11. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * (exp(x) ^ x)) * (((((1.875 / (x * x)) - -0.75) / (((x * x) * x) * x)) + (1.0 - (-0.5 / (x * x)))) / 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[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (((((1.875 / (x * x)) - -0.75) / (((x * x) * x) * x)) + (1.0 - (-0.5 / (x * x)))) / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

Alternative 5: 99.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * -(((-0.75 / (((x * x) * x) * x)) - (1.0 - (-0.5 / (x * x)))) / x);
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * -(((-0.75 / (((x * x) * x) * x)) - (1.0 - (-0.5 / (x * x)))) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * -(((-0.75 / (((x * x) * x) * x)) - (1.0 - (-0.5 / (x * x)))) / x);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{\frac{-3}{4}}{{x}^{4}} - \left(1 - \frac{\frac{-1}{2}}{x \cdot x}\right)}{x}\right) \]

    2. metadata-eval N/A

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

    3. pow-plus N/A

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{\frac{-3}{4}}{{x}^{3} \cdot x} - \left(1 - \frac{\frac{-1}{2}}{x \cdot x}\right)}{x}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{\frac{-3}{4}}{{x}^{3} \cdot x} - \left(1 - \frac{\frac{-1}{2}}{x \cdot x}\right)}{x}\right) \]

    5. pow3 N/A

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{\frac{-3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \left(1 - \frac{\frac{-1}{2}}{x \cdot x}\right)}{x}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{\frac{-3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \left(1 - \frac{\frac{-1}{2}}{x \cdot x}\right)}{x}\right) \]

    7. lift-*.f64 99.6

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \left(1 - \frac{-0.5}{x \cdot x}\right)}{x}\right) \]

12. Applied rewrites 99.6%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(-\frac{\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \left(1 - \frac{-0.5}{x \cdot x}\right)}{x}\right) \]
13. Add Preprocessing

Alternative 6: 99.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x);
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 - \frac{-0.5}{x \cdot x}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}}{\color{blue}{x}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. mult-flip N/A

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

   6. pow2 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 99.6

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 - \frac{-0.5}{x \cdot x}}{x} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 - \frac{-0.5}{x \cdot x}}{\color{blue}{x}} \]
10. Add Preprocessing

Alternative 8: 99.6% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 9: 1.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 1.8%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. pow3 N/A

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

    6. associate-/r* N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. metadata-eval N/A

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

    10. sqrt-div N/A

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

    11. unpow3 N/A

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

    12. pow2 N/A

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

    13. associate-/l/ N/A

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

11. Applied rewrites 1.8%

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

Alternative 10: 1.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 / N[(N[(N[(x * 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. 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%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 1.8%

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

    1. metadata-eval N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-*l/ N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. lower-/.f64 N/A

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

    9. metadata-eval 1.8

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. pow3 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f64 N/A

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

    16. pow3 N/A

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

    17. lift-*.f64 N/A

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

    18. lift-*.f64 1.8

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

11. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025128 
(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)))))))