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

Percentage Accurate: 100.0% → 99.9%

Time: 5.0s

Alternatives: 11

Speedup: N/A×

• 0.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

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 (/ 1.0 (fabs x)))
       (t_1 (* (* t_0 t_0) t_0))
       (t_2 (* (* t_1 t_0) t_0)))
  (*
   (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
   (+
    (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
    (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ t_1 := t\_0 \cdot x\\ t_2 := \frac{0.5}{x \cdot x} - -1\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|} \cdot \left(t\_2 - \frac{-0.75 \cdot \left(x \cdot x\right) - 1.875}{\left(t\_1 \cdot x\right) \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_2 - \left(\frac{-0.75}{t\_1} - \frac{1.875}{t\_0 \cdot t\_0}\right)\right) \cdot \left(1 + {x}^{2}\right)}{\left|x\right| \cdot 1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x))
       (t_1 (* t_0 x))
       (t_2 (- (/ 0.5 (* x x)) -1.0)))
  (if (<= x 1.35e+154)
    (*
     (/ (exp (* x x)) (* (sqrt PI) (fabs x)))
     (- t_2 (/ (- (* -0.75 (* x x)) 1.875) (* (* t_1 x) x))))
    (/
     (*
      (- t_2 (- (/ -0.75 t_1) (/ 1.875 (* t_0 t_0))))
      (+ 1.0 (pow x 2.0)))
     (* (fabs x) 1.772453850905516)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = t_0 * x;
	double t_2 = (0.5 / (x * x)) - -1.0;
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (exp((x * x)) / (sqrt(((double) M_PI)) * fabs(x))) * (t_2 - (((-0.75 * (x * x)) - 1.875) / ((t_1 * x) * x)));
	} else {
		tmp = ((t_2 - ((-0.75 / t_1) - (1.875 / (t_0 * t_0)))) * (1.0 + pow(x, 2.0))) / (fabs(x) * 1.772453850905516);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = t_0 * x;
	double t_2 = (0.5 / (x * x)) - -1.0;
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (Math.exp((x * x)) / (Math.sqrt(Math.PI) * Math.abs(x))) * (t_2 - (((-0.75 * (x * x)) - 1.875) / ((t_1 * x) * x)));
	} else {
		tmp = ((t_2 - ((-0.75 / t_1) - (1.875 / (t_0 * t_0)))) * (1.0 + Math.pow(x, 2.0))) / (Math.abs(x) * 1.772453850905516);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * x
	t_1 = t_0 * x
	t_2 = (0.5 / (x * x)) - -1.0
	tmp = 0
	if x <= 1.35e+154:
		tmp = (math.exp((x * x)) / (math.sqrt(math.pi) * math.fabs(x))) * (t_2 - (((-0.75 * (x * x)) - 1.875) / ((t_1 * x) * x)))
	else:
		tmp = ((t_2 - ((-0.75 / t_1) - (1.875 / (t_0 * t_0)))) * (1.0 + math.pow(x, 2.0))) / (math.fabs(x) * 1.772453850905516)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	t_1 = Float64(t_0 * x)
	t_2 = Float64(Float64(0.5 / Float64(x * x)) - -1.0)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(x * x)) / Float64(sqrt(pi) * abs(x))) * Float64(t_2 - Float64(Float64(Float64(-0.75 * Float64(x * x)) - 1.875) / Float64(Float64(t_1 * x) * x))));
	else
		tmp = Float64(Float64(Float64(t_2 - Float64(Float64(-0.75 / t_1) - Float64(1.875 / Float64(t_0 * t_0)))) * Float64(1.0 + (x ^ 2.0))) / Float64(abs(x) * 1.772453850905516));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * x;
	t_1 = t_0 * x;
	t_2 = (0.5 / (x * x)) - -1.0;
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (exp((x * x)) / (sqrt(pi) * abs(x))) * (t_2 - (((-0.75 * (x * x)) - 1.875) / ((t_1 * x) * x)));
	else
		tmp = ((t_2 - ((-0.75 / t_1) - (1.875 / (t_0 * t_0)))) * (1.0 + (x ^ 2.0))) / (abs(x) * 1.772453850905516);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, 1.35e+154], N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[(N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.875), $MachinePrecision] / N[(N[(t$95$1 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 - N[(N[(-0.75 / t$95$1), $MachinePrecision] - N[(1.875 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 1.772453850905516), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e154

   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%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 49.6%

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

   if 1.35e154 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

   5. Applied rewrites 99.9%

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

   6. Evaluated real constant 99.9%

      \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{1.772453850905516}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot \color{blue}{\left(1 + {x}^{2}\right)}}{\left|x\right| \cdot 1.772453850905516} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{\frac{15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot \left(1 + \color{blue}{{x}^{2}}\right)}{\left|x\right| \cdot \frac{7982422502469483}{4503599627370496}} \]

      2. lower-pow.f64 52.3%

         \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot \left(1 + {x}^{\color{blue}{2}}\right)}{\left|x\right| \cdot 1.772453850905516} \]

   9. Applied rewrites 52.3%

      \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot \color{blue}{\left(1 + {x}^{2}\right)}}{\left|x\right| \cdot 1.772453850905516} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(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)))) * exp(Float64(x * x))) / Float64(abs(x) * sqrt(pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

5. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{t\_0 \cdot x} - \frac{1.875}{t\_0 \cdot t\_0}\right)\right) \cdot e^{x \cdot x}}{\left|x\right| \cdot 1.772453850905516} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (/
   (*
    (-
     (- (/ 0.5 (* x x)) -1.0)
     (- (/ -0.75 (* t_0 x)) (/ 1.875 (* t_0 t_0))))
    (exp (* x x)))
   (* (fabs x) 1.772453850905516))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) * exp((x * x))) / (fabs(x) * 1.772453850905516);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x * x) * x
    code = ((((0.5d0 / (x * x)) - (-1.0d0)) - (((-0.75d0) / (t_0 * x)) - (1.875d0 / (t_0 * t_0)))) * exp((x * x))) / (abs(x) * 1.772453850905516d0)
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) * Math.exp((x * x))) / (Math.abs(x) * 1.772453850905516);
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) * math.exp((x * x))) / (math.fabs(x) * 1.772453850905516)

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(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)))) * exp(Float64(x * x))) / Float64(abs(x) * 1.772453850905516))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((((0.5 / (x * x)) - -1.0) - ((-0.75 / (t_0 * x)) - (1.875 / (t_0 * t_0)))) * exp((x * x))) / (abs(x) * 1.772453850905516);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(-0.75 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(1.875 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 1.772453850905516), $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%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

5. Applied rewrites 99.9%

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

6. Evaluated real constant 99.9%

   \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{1.772453850905516}} \]
7. Add Preprocessing

Alternative 4: 49.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 49.6%

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

Alternative 5: 49.6% accurate, 2.3× speedup

Math

\[\frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \frac{-0.75 \cdot \left(x \cdot x\right) - 1.875}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot 0.5641895835477563\right)}{\left|x\right|} \]

FPCore

(FPCore (x)
  :precision binary64
  (/
 (*
  (-
   (- (/ 0.5 (* x x)) -1.0)
   (/ (- (* -0.75 (* x x)) 1.875) (* (* (* (* (* x x) x) x) x) x)))
  (* (exp (* x x)) 0.5641895835477563))
 (fabs x)))

C

double code(double x) {
	return ((((0.5 / (x * x)) - -1.0) - (((-0.75 * (x * x)) - 1.875) / (((((x * x) * x) * x) * x) * x))) * (exp((x * x)) * 0.5641895835477563)) / fabs(x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((((0.5d0 / (x * x)) - (-1.0d0)) - ((((-0.75d0) * (x * x)) - 1.875d0) / (((((x * x) * x) * x) * x) * x))) * (exp((x * x)) * 0.5641895835477563d0)) / abs(x)
end function

Java

public static double code(double x) {
	return ((((0.5 / (x * x)) - -1.0) - (((-0.75 * (x * x)) - 1.875) / (((((x * x) * x) * x) * x) * x))) * (Math.exp((x * x)) * 0.5641895835477563)) / Math.abs(x);
}

Python

def code(x):
	return ((((0.5 / (x * x)) - -1.0) - (((-0.75 * (x * x)) - 1.875) / (((((x * x) * x) * x) * x) * x))) * (math.exp((x * x)) * 0.5641895835477563)) / math.fabs(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) - Float64(Float64(Float64(-0.75 * Float64(x * x)) - 1.875) / Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x))) * Float64(exp(Float64(x * x)) * 0.5641895835477563)) / abs(x))
end

MATLAB

function tmp = code(x)
	tmp = ((((0.5 / (x * x)) - -1.0) - (((-0.75 * (x * x)) - 1.875) / (((((x * x) * x) * x) * x) * x))) * (exp((x * x)) * 0.5641895835477563)) / abs(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.875), $MachinePrecision] / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

5. Applied rewrites 99.9%

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

6. Evaluated real constant 99.9%

   \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \cdot e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{1.772453850905516}} \]

8. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) - \frac{-0.75 \cdot \left(x \cdot x\right) - 1.875}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot 0.5641895835477563\right)}{\left|x\right|}} \]
9. Add Preprocessing

Alternative 6: 27.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ t_1 := \left(\frac{0.5}{x \cdot x} - -1\right) - \frac{-0.75}{t\_0}\\ \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right) \cdot t\_0 + \left|x\right| \cdot 1.875}{t\_0}}{t\_0} \cdot 1}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left|x\right| \cdot \left(\frac{1.875}{t\_0} + t\_1 \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{x \cdot x}}{x \cdot x}}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (* x x) x) x))
       (t_1 (- (- (/ 0.5 (* x x)) -1.0) (/ -0.75 t_0))))
  (if (<= x 1.15e+77)
    (/
     (*
      (/
       (/
        (+ (* (* (* t_1 (fabs x)) (* x x)) t_0) (* (fabs x) 1.875))
        t_0)
       t_0)
      1.0)
     (sqrt PI))
    (/
     (/
      (/
       (* (* (fabs x) (+ (/ 1.875 t_0) (* t_1 (* x x)))) 1.0)
       (* x x))
      (* x x))
     (sqrt PI)))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	double t_1 = ((0.5 / (x * x)) - -1.0) - (-0.75 / t_0);
	double tmp;
	if (x <= 1.15e+77) {
		tmp = (((((((t_1 * fabs(x)) * (x * x)) * t_0) + (fabs(x) * 1.875)) / t_0) / t_0) * 1.0) / sqrt(((double) M_PI));
	} else {
		tmp = ((((fabs(x) * ((1.875 / t_0) + (t_1 * (x * x)))) * 1.0) / (x * x)) / (x * x)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * x;
	double t_1 = ((0.5 / (x * x)) - -1.0) - (-0.75 / t_0);
	double tmp;
	if (x <= 1.15e+77) {
		tmp = (((((((t_1 * Math.abs(x)) * (x * x)) * t_0) + (Math.abs(x) * 1.875)) / t_0) / t_0) * 1.0) / Math.sqrt(Math.PI);
	} else {
		tmp = ((((Math.abs(x) * ((1.875 / t_0) + (t_1 * (x * x)))) * 1.0) / (x * x)) / (x * x)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((x * x) * x) * x
	t_1 = ((0.5 / (x * x)) - -1.0) - (-0.75 / t_0)
	tmp = 0
	if x <= 1.15e+77:
		tmp = (((((((t_1 * math.fabs(x)) * (x * x)) * t_0) + (math.fabs(x) * 1.875)) / t_0) / t_0) * 1.0) / math.sqrt(math.pi)
	else:
		tmp = ((((math.fabs(x) * ((1.875 / t_0) + (t_1 * (x * x)))) * 1.0) / (x * x)) / (x * x)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	t_1 = Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) - Float64(-0.75 / t_0))
	tmp = 0.0
	if (x <= 1.15e+77)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_1 * abs(x)) * Float64(x * x)) * t_0) + Float64(abs(x) * 1.875)) / t_0) / t_0) * 1.0) / sqrt(pi));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(abs(x) * Float64(Float64(1.875 / t_0) + Float64(t_1 * Float64(x * x)))) * 1.0) / Float64(x * x)) / Float64(x * x)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((x * x) * x) * x;
	t_1 = ((0.5 / (x * x)) - -1.0) - (-0.75 / t_0);
	tmp = 0.0;
	if (x <= 1.15e+77)
		tmp = (((((((t_1 * abs(x)) * (x * x)) * t_0) + (abs(x) * 1.875)) / t_0) / t_0) * 1.0) / sqrt(pi);
	else
		tmp = ((((abs(x) * ((1.875 / t_0) + (t_1 * (x * x)))) * 1.0) / (x * x)) / (x * x)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(-0.75 / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.15e+77], N[(N[(N[(N[(N[(N[(N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 1.875), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(1.875 / t$95$0), $MachinePrecision] + N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e77

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 2.3%

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

   8. Applied rewrites 0.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 11.1%

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

   if 1.15e77 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 2.3%

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

   8. Applied rewrites 0.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

   10. Applied rewrites 17.2%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left|x\right| \cdot \left(\frac{1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} + \left(\left(\frac{0.5}{x \cdot x} - -1\right) - \frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{x \cdot x}}{x \cdot x}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 17.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 2.3%

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

8. Applied rewrites 0.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

10. Applied rewrites 17.2%

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

Alternative 8: 2.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 2.3%

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

8. Applied rewrites 2.3%

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

Alternative 9: 2.2% accurate, 4.3× speedup

Math

\[\frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) \cdot \left|x\right| - \frac{-1.875}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \cdot \left|x\right|\right) \cdot 1}{\left(x \cdot x\right) \cdot \sqrt{\pi}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/
 (*
  (-
   (* (- (/ 0.5 (* x x)) -1.0) (fabs x))
   (* (/ -1.875 (* (* (* (* (* x x) x) x) x) x)) (fabs x)))
  1.0)
 (* (* x x) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 2.3%

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

8. Applied rewrites 1.4%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{\left(\left(\frac{0.5}{x \cdot x} - -1\right) \cdot \left|x\right| - \frac{\color{blue}{\frac{-15}{8}}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \cdot \left|x\right|\right) \cdot 1}{\left(x \cdot x\right) \cdot \sqrt{\pi}} \]
10. Step-by-step derivation

11. Applied rewrites 2.2%

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

Alternative 10: 1.7% accurate, 7.2× speedup

Math

\[\frac{\frac{1.875}{\sqrt{\pi} \cdot \left|x\right|}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (/ 1.875 (* (sqrt PI) (fabs x))) (* (* (* (* (* x x) x) x) x) x)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.875 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $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%

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

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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 1.7%

      \[\leadsto \frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

6. Applied rewrites 1.7%

   \[\leadsto \color{blue}{\frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
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

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

8. Applied rewrites 1.7%

   \[\leadsto \frac{\frac{1.875}{\sqrt{\pi} \cdot \left|x\right|}}{\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}} \]
9. Add Preprocessing

Alternative 11: 1.7% accurate, 7.9× speedup

Math

\[\frac{1.875}{\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(\sqrt{\pi} \cdot \left|x\right|\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 1.875 (* (* x x) (* x (* (* (sqrt PI) (fabs x)) (* (* x x) x))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.875 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 1.7%

      \[\leadsto \frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)} \]

6. Applied rewrites 1.7%

   \[\leadsto \color{blue}{\frac{1.875}{{x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow3 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 1.7%

       \[\leadsto \frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \sqrt{\pi}\right)} \]

8. Applied rewrites 1.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 1.7%

       \[\leadsto \frac{1.875}{\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(\sqrt{\pi} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right)} \]

10. Applied rewrites 1.7%

    \[\leadsto \frac{1.875}{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\left(\sqrt{\pi} \cdot \left|x\right|\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))