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

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 11

Speedup: 1.4×

• 0.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq \frac{1}{2}\]

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 1 (fabs x)))
       (t_1 (* (* t_0 t_0) t_0))
       (t_2 (* (* t_1 t_0) t_0)))
  (*
   (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x))))
   (+
    (+ (+ t_0 (* (/ 1 2) t_1)) (* (/ 3 4) t_2))
    (* (/ 15 8) (* (* 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 / 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 / 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 / 2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3 / 4), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15 / 8), $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

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 1 (fabs x)))
       (t_1 (* (* t_0 t_0) t_0))
       (t_2 (* (* t_1 t_0) t_0)))
  (*
   (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x))))
   (+
    (+ (+ t_0 (* (/ 1 2) t_1)) (* (/ 3 4) t_2))
    (* (/ 15 8) (* (* 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 / 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 / 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 / 2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3 / 4), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15 / 8), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (* (exp (* (log (sqrt PI)) -1)) (pow (exp x) x))
   (/
    (-
     (- (/ 1/2 (* x x)) -1)
     (- (/ -3/4 (* t_0 x)) (/ 15/8 (* t_0 t_0))))
    (fabs x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[Exp[N[(N[Log[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision] * -1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(15/8 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. lower-unsound-exp.f64 N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-log.f64 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (sqrt (/ (pow (exp x) (+ x x)) PI))
   (/
    (-
     (- (/ 1/2 (* x x)) -1)
     (- (/ -3/4 (* t_0 x)) (/ 15/8 (* t_0 t_0))))
    (fabs x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Sqrt[N[(N[Power[N[Exp[x], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(15/8 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-lft-sqr N/A

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

   4. lift-*.f64 N/A

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

   5. pow-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. pow-prod-up N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-+.f64 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (sqrt (/ (exp (* (* x x) 2)) PI))
   (-
    (/ (- (/ 3/4 (* t_0 x)) (/ -1/2 (* x x))) (fabs x))
    (- (/ -15/8 (* (* t_0 t_0) (fabs x))) (/ 1 (fabs x)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return sqrt((exp(((x * x) * 2.0)) / ((double) M_PI))) * ((((0.75 / (t_0 * x)) - (-0.5 / (x * x))) / fabs(x)) - ((-1.875 / ((t_0 * t_0) * fabs(x))) - (1.0 / fabs(x))));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return Math.sqrt((Math.exp(((x * x) * 2.0)) / Math.PI)) * ((((0.75 / (t_0 * x)) - (-0.5 / (x * x))) / Math.abs(x)) - ((-1.875 / ((t_0 * t_0) * Math.abs(x))) - (1.0 / Math.abs(x))));
}

Python

def code(x):
	t_0 = (x * x) * x
	return math.sqrt((math.exp(((x * x) * 2.0)) / math.pi)) * ((((0.75 / (t_0 * x)) - (-0.5 / (x * x))) / math.fabs(x)) - ((-1.875 / ((t_0 * t_0) * math.fabs(x))) - (1.0 / math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(sqrt(Float64(exp(Float64(Float64(x * x) * 2.0)) / pi)) * Float64(Float64(Float64(Float64(0.75 / Float64(t_0 * x)) - Float64(-0.5 / Float64(x * x))) / abs(x)) - Float64(Float64(-1.875 / Float64(Float64(t_0 * t_0) * abs(x))) - Float64(1.0 / abs(x)))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = sqrt((exp(((x * x) * 2.0)) / pi)) * ((((0.75 / (t_0 * x)) - (-0.5 / (x * x))) / abs(x)) - ((-1.875 / ((t_0 * t_0) * abs(x))) - (1.0 / abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Sqrt[N[(N[Exp[N[(N[(x * x), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(-1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(N[(-15/8 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (sqrt (/ (exp (* (* x x) 2)) PI))
   (/
    (-
     (- (/ 1/2 (* x x)) -1)
     (- (/ -3/4 (* t_0 x)) (/ 15/8 (* t_0 t_0))))
    (fabs x)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return sqrt((exp(((x * x) * 2.0)) / ((double) M_PI))) * ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) / fabs(x));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return Math.sqrt((Math.exp(((x * x) * 2.0)) / Math.PI)) * ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) / Math.abs(x));
}

Python

def code(x):
	t_0 = (x * x) * x
	return math.sqrt((math.exp(((x * x) * 2.0)) / math.pi)) * ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) / math.fabs(x))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(sqrt(Float64(exp(Float64(Float64(x * x) * 2.0)) / pi)) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) - Float64(Float64(-0.75 / Float64(t_0 * x)) - Float64(1.875 / Float64(t_0 * t_0)))) / abs(x)))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = sqrt((exp(((x * x) * 2.0)) / pi)) * ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) / abs(x));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Sqrt[N[(N[Exp[N[(N[(x * x), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(15/8 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (/
   (*
    (/ (exp (* x x)) (sqrt PI))
    (-
     (- (/ 1/2 (* x x)) -1)
     (- (/ -3/4 (* t_0 x)) (/ 15/8 (* t_0 t_0)))))
   (fabs x))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(15/8 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 100.0%

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

Alternative 6: 99.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (sqrt (/ (exp (* (* x x) 2)) PI))
 (/
  (- (/ 1/2 (* x x)) (- -1 (/ 15/8 (* (* (* (* (* x x) x) x) x) x))))
  (fabs x))))

C

double code(double x) {
	return sqrt((exp(((x * x) * 2.0)) / ((double) M_PI))) * (((0.5 / (x * x)) - (-1.0 - (1.875 / (((((x * x) * x) * x) * x) * x)))) / fabs(x));
}

Java

public static double code(double x) {
	return Math.sqrt((Math.exp(((x * x) * 2.0)) / Math.PI)) * (((0.5 / (x * x)) - (-1.0 - (1.875 / (((((x * x) * x) * x) * x) * x)))) / Math.abs(x));
}

Python

def code(x):
	return math.sqrt((math.exp(((x * x) * 2.0)) / math.pi)) * (((0.5 / (x * x)) - (-1.0 - (1.875 / (((((x * x) * x) * x) * x) * x)))) / math.fabs(x))

Julia

function code(x)
	return Float64(sqrt(Float64(exp(Float64(Float64(x * x) * 2.0)) / pi)) * Float64(Float64(Float64(0.5 / Float64(x * x)) - Float64(-1.0 - Float64(1.875 / Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x)))) / abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((exp(((x * x) * 2.0)) / pi)) * (((0.5 / (x * x)) - (-1.0 - (1.875 / (((((x * x) * x) * x) * x) * x)))) / abs(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(N[Exp[N[(N[(x * x), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(-1 - N[(15/8 / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 49.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.6%

    \[\leadsto \sqrt{\frac{e^{\left(x \cdot x\right) \cdot 2}}{\pi}} \cdot \frac{\frac{\frac{1}{2}}{x \cdot x} - \left(-1 - \frac{\color{blue}{\frac{15}{8}}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right)}{\left|x\right|} \]
11. Add Preprocessing

Alternative 7: 99.6% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (sqrt (/ (exp (* (* x x) 2)) PI))
 (/ (- (/ 1/2 (* x x)) -1) (fabs x))))

C

double code(double x) {
	return sqrt((exp(((x * x) * 2.0)) / ((double) M_PI))) * (((0.5 / (x * x)) - -1.0) / fabs(x));
}

Java

public static double code(double x) {
	return Math.sqrt((Math.exp(((x * x) * 2.0)) / Math.PI)) * (((0.5 / (x * x)) - -1.0) / Math.abs(x));
}

Python

def code(x):
	return math.sqrt((math.exp(((x * x) * 2.0)) / math.pi)) * (((0.5 / (x * x)) - -1.0) / math.fabs(x))

Julia

function code(x)
	return Float64(sqrt(Float64(exp(Float64(Float64(x * x) * 2.0)) / pi)) * Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) / abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((exp(((x * x) * 2.0)) / pi)) * (((0.5 / (x * x)) - -1.0) / abs(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(N[Exp[N[(N[(x * x), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 49.2%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.6%

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

Alternative 8: 99.6% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt (/ (exp (* (* x x) 2)) PI)) (/ 1 (fabs x))))

C

double code(double x) {
	return sqrt((exp(((x * x) * 2.0)) / ((double) M_PI))) * (1.0 / fabs(x));
}

Java

public static double code(double x) {
	return Math.sqrt((Math.exp(((x * x) * 2.0)) / Math.PI)) * (1.0 / Math.abs(x));
}

Python

def code(x):
	return math.sqrt((math.exp(((x * x) * 2.0)) / math.pi)) * (1.0 / math.fabs(x))

Julia

function code(x)
	return Float64(sqrt(Float64(exp(Float64(Float64(x * x) * 2.0)) / pi)) * Float64(1.0 / abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((exp(((x * x) * 2.0)) / pi)) * (1.0 / abs(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(N[Exp[N[(N[(x * x), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(1 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fabs.f64 99.6%

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

8. Applied rewrites 99.6%

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

Alternative 9: 2.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \sqrt{\frac{1}{\pi}} \cdot \frac{\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-3}{4}}{t\_0 \cdot x} - \frac{\frac{15}{8}}{t\_0 \cdot t\_0}\right)}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (sqrt (/ 1 PI))
   (/
    (-
     (- (/ 1/2 (* x x)) -1)
     (- (/ -3/4 (* t_0 x)) (/ 15/8 (* t_0 t_0))))
    (fabs x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Sqrt[N[(1 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(1/2 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-3/4 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(15/8 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-fabs N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-PI.f64 2.3%

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

8. Applied rewrites 2.3%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \frac{\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{\frac{15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)}{\left|x\right|} \]
9. Add Preprocessing

Alternative 10: 1.7% accurate, 7.2× speedup

Math

\[\frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (/ 15/8 (* (fabs x) (sqrt PI))) (* (* (* (* x x) x) (* x x)) x)))

C

double code(double x) {
	return (1.875 / (fabs(x) * sqrt(((double) M_PI)))) / ((((x * x) * x) * (x * x)) * x);
}

Java

public static double code(double x) {
	return (1.875 / (Math.abs(x) * Math.sqrt(Math.PI))) / ((((x * x) * x) * (x * x)) * x);
}

Python

def code(x):
	return (1.875 / (math.fabs(x) * math.sqrt(math.pi))) / ((((x * x) * x) * (x * x)) * x)

Julia

function code(x)
	return Float64(Float64(1.875 / Float64(abs(x) * sqrt(pi))) / Float64(Float64(Float64(Float64(x * x) * x) * Float64(x * x)) * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.875 / (abs(x) * sqrt(pi))) / ((((x * x) * x) * (x * x)) * x);
end

Wolfram

code[x_] := N[(N[(15/8 / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

   7. lower-fabs.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   9. lower-PI.f64 1.7%

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

6. Applied rewrites 1.7%

   \[\leadsto \color{blue}{\frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \color{blue}{{x}^{6}}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot {x}^{\color{blue}{6}}} \]

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow3 N/A

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

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right)} \]

   9. lift-*.f64 N/A

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

   10. pow3 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)} \]

   14. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}} \]

   16. lower-/.f64 1.7%

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)} \]

   19. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{x}} \]

8. Applied rewrites 1.7%

   \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]

   5. lower-*.f64 1.7%

      \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]

10. Applied rewrites 1.7%

    \[\leadsto \frac{\frac{\frac{15}{8}}{\left|x\right| \cdot \sqrt{\pi}}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]
11. Add Preprocessing

Alternative 11: 1.7% accurate, 7.9× speedup

Math

\[\frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 15/8 (* (* (fabs x) (* (* (* (* x x) (* x x)) x) x)) (sqrt PI))))

C

double code(double x) {
	return 1.875 / ((fabs(x) * ((((x * x) * (x * x)) * x) * x)) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 1.875 / ((Math.abs(x) * ((((x * x) * (x * x)) * x) * x)) * Math.sqrt(Math.PI));
}

Python

def code(x):
	return 1.875 / ((math.fabs(x) * ((((x * x) * (x * x)) * x) * x)) * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(1.875 / Float64(Float64(abs(x) * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * x) * x)) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 1.875 / ((abs(x) * ((((x * x) * (x * x)) * x) * x)) * sqrt(pi));
end

Wolfram

code[x_] := N[(15/8 / N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{15}{8}}{\color{blue}{{x}^{6}} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

   7. lower-fabs.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   9. lower-PI.f64 1.7%

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

6. Applied rewrites 1.7%

   \[\leadsto \color{blue}{\frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{{x}^{6} \cdot \left(\left|x\right| \cdot \color{blue}{\sqrt{\pi}}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{15}{8}}{\left({x}^{6} \cdot \left|x\right|\right) \cdot \color{blue}{\sqrt{\pi}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left({x}^{6} \cdot \left|x\right|\right) \cdot \color{blue}{\sqrt{\pi}}} \]

8. Applied rewrites 1.7%

   \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\sqrt{\pi}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

   7. lower-*.f64 1.7%

      \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]

10. Applied rewrites 1.7%

    \[\leadsto \frac{\frac{15}{8}}{\left(\left|x\right| \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\right) \cdot \sqrt{\pi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 1/2)
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))