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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 11

Speedup: 2.7×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.8× speedup

Math

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

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(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(Float64(Float64(x * x) * x) * x))) - 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[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]) - 1.0), $MachinePrecision] - 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. 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%

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

5. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

10. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.1× speedup

Math

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

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(-Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(Float64(x * x) * Float64(x * x)))) - 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[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - 1.0), $MachinePrecision] - 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. 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%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 100.0%

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

Alternative 5: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-neg.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-+.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. div-sub N/A

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

    10. lower--.f64 N/A

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

11. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. pow2 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 8: 99.5% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. lift-/.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 9: 99.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. associate-/l/ N/A

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

   6. pow2 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\color{blue}{\pi}} \cdot x} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{{x}^{2}}}{x \cdot \color{blue}{\sqrt{\pi}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{x \cdot \color{blue}{\sqrt{\pi}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{x \cdot \sqrt{\pi}}} \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{e^{{x}^{2}}}{\color{blue}{x} \cdot \sqrt{\pi}} \]

   11. pow2 N/A

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

   12. lift-*.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 10: 50.7% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. distribute-rgt1-in N/A

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

    2. +-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. pow2 N/A

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

    6. lower-fma.f64 N/A

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

    7. sqrt-div N/A

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

    8. metadata-eval N/A

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

    9. lift-sqrt.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. lower-/.f64 50.7

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

11. Applied rewrites 50.7%

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

Alternative 11: 2.3% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. sqrt-div N/A

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

    2. metadata-eval N/A

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

    3. lift-sqrt.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. frac-times N/A

       \[\leadsto \frac{1 \cdot 1}{x \cdot \color{blue}{\sqrt{\pi}}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{x \cdot \sqrt{\color{blue}{\pi}}} \]

    7. lift-*.f64 N/A

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

    8. lower-/.f64 2.3

       \[\leadsto \frac{1}{x \cdot \color{blue}{\sqrt{\pi}}} \]

11. Applied rewrites 2.3%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \sqrt{\pi}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(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)))))))