Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 13

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cosh x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (cosh x)))

C

double code(double x) {
	return 1.0 / cosh(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / cosh(x)
end function

Java

public static double code(double x) {
	return 1.0 / Math.cosh(x);
}

Python

def code(x):
	return 1.0 / math.cosh(x)

Julia

function code(x)
	return Float64(1.0 / cosh(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / cosh(x);
end

Wolfram

code[x_] := N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ t_2 := x \cdot x + -2\\ t_3 := x \cdot \left(x \cdot t\_2\right)\\ t_4 := \left(x \cdot x\right) \cdot t\_2\\ t_5 := 0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{2 \cdot \left(64 + t\_4 \cdot \left(t\_4 \cdot t\_4\right)\right)}{\frac{\left(64 - x \cdot \left(\left(x \cdot t\_0\right) \cdot t\_1\right)\right) \cdot \left(16 + t\_3 \cdot \left(t\_3 + -4\right)\right)}{8 - t\_1}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{1 - t\_0 \cdot \left(t\_5 \cdot t\_5\right)} \cdot \left(1 - \left(x \cdot x\right) \cdot t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x))))
        (t_1 (* (* x x) t_0))
        (t_2 (+ (* x x) -2.0))
        (t_3 (* x (* x t_2)))
        (t_4 (* (* x x) t_2))
        (t_5
         (+
          0.5
          (*
           x
           (* x (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))))))
   (if (<= x 4.5e+25)
     (/
      (* 2.0 (+ 64.0 (* t_4 (* t_4 t_4))))
      (/
       (* (- 64.0 (* x (* (* x t_0) t_1))) (+ 16.0 (* t_3 (+ t_3 -4.0))))
       (- 8.0 t_1)))
     (if (<= x 5e+48)
       (* (/ 1.0 (- 1.0 (* t_0 (* t_5 t_5)))) (- 1.0 (* (* x x) t_5)))
       (/ 720.0 t_1)))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = (x * x) + -2.0;
	double t_3 = x * (x * t_2);
	double t_4 = (x * x) * t_2;
	double t_5 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	double tmp;
	if (x <= 4.5e+25) {
		tmp = (2.0 * (64.0 + (t_4 * (t_4 * t_4)))) / (((64.0 - (x * ((x * t_0) * t_1))) * (16.0 + (t_3 * (t_3 + -4.0)))) / (8.0 - t_1));
	} else if (x <= 5e+48) {
		tmp = (1.0 / (1.0 - (t_0 * (t_5 * t_5)))) * (1.0 - ((x * x) * t_5));
	} else {
		tmp = 720.0 / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = (x * x) * t_0
    t_2 = (x * x) + (-2.0d0)
    t_3 = x * (x * t_2)
    t_4 = (x * x) * t_2
    t_5 = 0.5d0 + (x * (x * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0)))))
    if (x <= 4.5d+25) then
        tmp = (2.0d0 * (64.0d0 + (t_4 * (t_4 * t_4)))) / (((64.0d0 - (x * ((x * t_0) * t_1))) * (16.0d0 + (t_3 * (t_3 + (-4.0d0))))) / (8.0d0 - t_1))
    else if (x <= 5d+48) then
        tmp = (1.0d0 / (1.0d0 - (t_0 * (t_5 * t_5)))) * (1.0d0 - ((x * x) * t_5))
    else
        tmp = 720.0d0 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = (x * x) + -2.0;
	double t_3 = x * (x * t_2);
	double t_4 = (x * x) * t_2;
	double t_5 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	double tmp;
	if (x <= 4.5e+25) {
		tmp = (2.0 * (64.0 + (t_4 * (t_4 * t_4)))) / (((64.0 - (x * ((x * t_0) * t_1))) * (16.0 + (t_3 * (t_3 + -4.0)))) / (8.0 - t_1));
	} else if (x <= 5e+48) {
		tmp = (1.0 / (1.0 - (t_0 * (t_5 * t_5)))) * (1.0 - ((x * x) * t_5));
	} else {
		tmp = 720.0 / t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = (x * x) * t_0
	t_2 = (x * x) + -2.0
	t_3 = x * (x * t_2)
	t_4 = (x * x) * t_2
	t_5 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))
	tmp = 0
	if x <= 4.5e+25:
		tmp = (2.0 * (64.0 + (t_4 * (t_4 * t_4)))) / (((64.0 - (x * ((x * t_0) * t_1))) * (16.0 + (t_3 * (t_3 + -4.0)))) / (8.0 - t_1))
	elif x <= 5e+48:
		tmp = (1.0 / (1.0 - (t_0 * (t_5 * t_5)))) * (1.0 - ((x * x) * t_5))
	else:
		tmp = 720.0 / t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(Float64(x * x) * t_0)
	t_2 = Float64(Float64(x * x) + -2.0)
	t_3 = Float64(x * Float64(x * t_2))
	t_4 = Float64(Float64(x * x) * t_2)
	t_5 = Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889))))))
	tmp = 0.0
	if (x <= 4.5e+25)
		tmp = Float64(Float64(2.0 * Float64(64.0 + Float64(t_4 * Float64(t_4 * t_4)))) / Float64(Float64(Float64(64.0 - Float64(x * Float64(Float64(x * t_0) * t_1))) * Float64(16.0 + Float64(t_3 * Float64(t_3 + -4.0)))) / Float64(8.0 - t_1)));
	elseif (x <= 5e+48)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(t_0 * Float64(t_5 * t_5)))) * Float64(1.0 - Float64(Float64(x * x) * t_5)));
	else
		tmp = Float64(720.0 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	t_1 = (x * x) * t_0;
	t_2 = (x * x) + -2.0;
	t_3 = x * (x * t_2);
	t_4 = (x * x) * t_2;
	t_5 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	tmp = 0.0;
	if (x <= 4.5e+25)
		tmp = (2.0 * (64.0 + (t_4 * (t_4 * t_4)))) / (((64.0 - (x * ((x * t_0) * t_1))) * (16.0 + (t_3 * (t_3 + -4.0)))) / (8.0 - t_1));
	elseif (x <= 5e+48)
		tmp = (1.0 / (1.0 - (t_0 * (t_5 * t_5)))) * (1.0 - ((x * x) * t_5));
	else
		tmp = 720.0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.5e+25], N[(N[(2.0 * N[(64.0 + N[(t$95$4 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(64.0 - N[(x * N[(N[(x * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(16.0 + N[(t$95$3 * N[(t$95$3 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(8.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+48], N[(N[(1.0 / N[(1.0 - N[(t$95$0 * N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(x * x), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.5000000000000003e25

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      3. *-lowering-*.f64 77.0%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 77.0%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(64 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(16 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) - 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(64, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right)\right), \left(\frac{8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(\color{blue}{16} + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) - 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(64, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right)\right), \left(\frac{\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(16 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) - 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)}{\color{blue}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(64, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(16 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) - 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)\right), \color{blue}{\left(8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right)\right) \]

   9. Applied egg-rr 66.5%

      \[\leadsto \frac{2 \cdot \left(64 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)}{\color{blue}{\frac{\left(64 - x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \cdot \left(16 + \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right) + -4\right)\right)}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}} \]

   if 4.5000000000000003e25 < x < 4.99999999999999973e48

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 7.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)} \cdot \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}\right), \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]

   if 4.99999999999999973e48 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)}\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{720}{{x}^{6}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \color{blue}{\left({x}^{6}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right)\right) \]

       4. cube-prod N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left({x}^{2}\right)}^{3}\right)\right) \]

       6. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := 0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{1 - t\_0 \cdot \left(t\_1 \cdot t\_1\right)} \cdot \left(1 - \left(x \cdot x\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x))))
        (t_1
         (+
          0.5
          (*
           x
           (* x (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))))))
   (if (<= x 5e+48)
     (* (/ 1.0 (- 1.0 (* t_0 (* t_1 t_1)))) (- 1.0 (* (* x x) t_1)))
     (/ 720.0 (* (* x x) t_0)))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	double tmp;
	if (x <= 5e+48) {
		tmp = (1.0 / (1.0 - (t_0 * (t_1 * t_1)))) * (1.0 - ((x * x) * t_1));
	} else {
		tmp = 720.0 / ((x * x) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = 0.5d0 + (x * (x * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0)))))
    if (x <= 5d+48) then
        tmp = (1.0d0 / (1.0d0 - (t_0 * (t_1 * t_1)))) * (1.0d0 - ((x * x) * t_1))
    else
        tmp = 720.0d0 / ((x * x) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	double tmp;
	if (x <= 5e+48) {
		tmp = (1.0 / (1.0 - (t_0 * (t_1 * t_1)))) * (1.0 - ((x * x) * t_1));
	} else {
		tmp = 720.0 / ((x * x) * t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))
	tmp = 0
	if x <= 5e+48:
		tmp = (1.0 / (1.0 - (t_0 * (t_1 * t_1)))) * (1.0 - ((x * x) * t_1))
	else:
		tmp = 720.0 / ((x * x) * t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889))))))
	tmp = 0.0
	if (x <= 5e+48)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(t_0 * Float64(t_1 * t_1)))) * Float64(1.0 - Float64(Float64(x * x) * t_1)));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	t_1 = 0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889)))));
	tmp = 0.0;
	if (x <= 5e+48)
		tmp = (1.0 / (1.0 - (t_0 * (t_1 * t_1)))) * (1.0 - ((x * x) * t_1));
	else
		tmp = 720.0 / ((x * x) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+48], N[(N[(1.0 / N[(1.0 - N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999973e48

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 87.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.3%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)} \cdot \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}\right), \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)}\right) \]

   9. Applied egg-rr 64.9%

      \[\leadsto \color{blue}{\frac{1}{1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot \left(1 - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]

   if 4.99999999999999973e48 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)}\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{720}{{x}^{6}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \color{blue}{\left({x}^{6}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right)\right) \]

       4. cube-prod N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left({x}^{2}\right)}^{3}\right)\right) \]

       6. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := 0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - t\_1 \cdot \left(t\_0 \cdot t\_2\right)\right)}{0.5 - x \cdot t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))
        (t_2 (* x t_1)))
   (if (<= x 1.5e+77)
     (/
      1.0
      (+ 1.0 (/ (* (* x x) (- 0.25 (* t_1 (* t_0 t_2)))) (- 0.5 (* x t_2)))))
     (/ 24.0 (* x t_0)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double t_2 = x * t_1;
	double tmp;
	if (x <= 1.5e+77) {
		tmp = 1.0 / (1.0 + (((x * x) * (0.25 - (t_1 * (t_0 * t_2)))) / (0.5 - (x * t_2))));
	} else {
		tmp = 24.0 / (x * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = 0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))
    t_2 = x * t_1
    if (x <= 1.5d+77) then
        tmp = 1.0d0 / (1.0d0 + (((x * x) * (0.25d0 - (t_1 * (t_0 * t_2)))) / (0.5d0 - (x * t_2))))
    else
        tmp = 24.0d0 / (x * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double t_2 = x * t_1;
	double tmp;
	if (x <= 1.5e+77) {
		tmp = 1.0 / (1.0 + (((x * x) * (0.25 - (t_1 * (t_0 * t_2)))) / (0.5 - (x * t_2))));
	} else {
		tmp = 24.0 / (x * t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = 0.041666666666666664 + (x * (x * 0.001388888888888889))
	t_2 = x * t_1
	tmp = 0
	if x <= 1.5e+77:
		tmp = 1.0 / (1.0 + (((x * x) * (0.25 - (t_1 * (t_0 * t_2)))) / (0.5 - (x * t_2))))
	else:
		tmp = 24.0 / (x * t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (x <= 1.5e+77)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.25 - Float64(t_1 * Float64(t_0 * t_2)))) / Float64(0.5 - Float64(x * t_2)))));
	else
		tmp = Float64(24.0 / Float64(x * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	t_2 = x * t_1;
	tmp = 0.0;
	if (x <= 1.5e+77)
		tmp = 1.0 / (1.0 + (((x * x) * (0.25 - (t_1 * (t_0 * t_2)))) / (0.5 - (x * t_2))));
	else
		tmp = 24.0 / (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[x, 1.5e+77], N[(1.0 / N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.25 - N[(t$95$1 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e77

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 87.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.8%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 66.7%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}}} \]

   if 1.4999999999999999e77 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

   5. Simplified 98.5%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)}{0.5 - x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   1.0
   (*
    (* x x)
    (+
     0.5
     (* x (* x (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))))

C

double code(double x) {
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
}

Python

def code(x):
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 90.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 90.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]
8. Add Preprocessing

Alternative 6: 92.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+ 1.0 (* (* x x) (+ 0.5 (* x (* x (* (* x x) 0.001388888888888889))))))))

C

double code(double x) {
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + ((x * x) * (0.5d0 + (x * (x * ((x * x) * 0.001388888888888889d0))))))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
}

Python

def code(x):
	return 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(Float64(x * x) * 0.001388888888888889)))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 90.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 90.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{3}\right)}\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 90.4%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 90.4%

    \[\leadsto \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}\right)} \]
11. Add Preprocessing

Alternative 7: 71.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.3)
   (+ 1.0 (* (* x x) (+ -0.5 (* (* x x) 0.20833333333333334))))
   (/ 720.0 (* (* x x) (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.3) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.3d0) then
        tmp = 1.0d0 + ((x * x) * ((-0.5d0) + ((x * x) * 0.20833333333333334d0)))
    else
        tmp = 720.0d0 / ((x * x) * (x * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.3) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.3:
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)))
	else:
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.3)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * 0.20833333333333334))));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.3)
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	else
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.3], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{5}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{5}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{5}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{5}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{\frac{5}{24} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{5}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{5}{24}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{5}{24}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{5}{24}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 66.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{5}{24}\right)\right)\right)\right) \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)} \]

   if 2.2999999999999998 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

      3. cosh-def N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

      4. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 80.6%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 80.6%

       \[\leadsto \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)}\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{720}{{x}^{6}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \color{blue}{\left({x}^{6}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right)\right) \]

       4. cube-prod N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({\left({x}^{2}\right)}^{3}\right)\right) \]

       6. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 80.6%

           \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 80.6%

       \[\leadsto \color{blue}{\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (+ 1.0 (* (* x x) (+ -0.5 (* (* x x) 0.20833333333333334))))
   (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.9d0) then
        tmp = 1.0d0 + ((x * x) * ((-0.5d0) + ((x * x) * 0.20833333333333334d0)))
    else
        tmp = 24.0d0 / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)))
	else:
		tmp = 24.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * 0.20833333333333334))));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	else
		tmp = 24.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{5}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{5}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{5}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{5}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{\frac{5}{24} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{5}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{5}{24}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{5}{24}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{5}{24}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 66.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{5}{24}\right)\right)\right)\right) \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)} \]

   if 1.8999999999999999 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

   5. Simplified 71.2%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 72.3%

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 92.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.001388888888888889} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ 1.0 (* (* (* x x) (* x (* x (* x x)))) 0.001388888888888889))))

C

double code(double x) {
	return 1.0 / (1.0 + (((x * x) * (x * (x * (x * x)))) * 0.001388888888888889));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (((x * x) * (x * (x * (x * x)))) * 0.001388888888888889d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + (((x * x) * (x * (x * (x * x)))) * 0.001388888888888889));
}

Python

def code(x):
	return 1.0 / (1.0 + (((x * x) * (x * (x * (x * x)))) * 0.001388888888888889))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))) * 0.001388888888888889)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (((x * x) * (x * (x * (x * x)))) * 0.001388888888888889));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}\right)}\right) \]

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 90.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 90.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 90.4%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 90.4%

    \[\leadsto \frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)}\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{720} \cdot {x}^{6}\right)}\right)\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({x}^{6} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{6}\right), \color{blue}{\frac{1}{720}}\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 3\right)}\right), \frac{1}{720}\right)\right)\right) \]

    4. pow-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{3} \cdot {x}^{3}\right), \frac{1}{720}\right)\right)\right) \]

    5. cube-prod N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{3}\right), \frac{1}{720}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{3}\right), \frac{1}{720}\right)\right)\right) \]

    7. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), \frac{1}{720}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot {x}^{2}\right)\right), \frac{1}{720}\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot {x}^{2}\right)\right), \frac{1}{720}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot {x}^{2}\right)\right), \frac{1}{720}\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right), \frac{1}{720}\right)\right)\right) \]

    12. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    14. cube-mult N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot {x}^{3}\right)\right), \frac{1}{720}\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left({x}^{3}\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    16. cube-mult N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    18. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    19. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

    20. *-lowering-*.f64 90.2%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{1}{720}\right)\right)\right) \]

13. Simplified 90.2%

    \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.001388888888888889}} \]
14. Add Preprocessing

Alternative 10: 81.8% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.7) (/ 2.0 (+ 2.0 (* x x))) (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.7d0) then
        tmp = 2.0d0 / (2.0d0 + (x * x))
    else
        tmp = 24.0d0 / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.7:
		tmp = 2.0 / (2.0 + (x * x))
	else:
		tmp = 24.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.7)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * x)));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.7)
		tmp = 2.0 / (2.0 + (x * x));
	else
		tmp = 24.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.7], N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      3. *-lowering-*.f64 81.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 81.5%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   if 3.7000000000000002 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{12}\right)\right)\right)\right)\right) \]

   5. Simplified 71.2%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 72.3%

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.4% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 67.2%

      \[\leadsto \color{blue}{1} \]

   if 1.3999999999999999 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      3. *-lowering-*.f64 46.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 46.7%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 46.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{2}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.8% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ 2.0 (* x x))))

C

double code(double x) {
	return 2.0 / (2.0 + (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + (x * x))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * x));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * x))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + (x * x));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   3. *-lowering-*.f64 70.9%

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 70.9%

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]
6. Add Preprocessing

Alternative 13: 51.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 47.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))