Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 14

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: 75.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\\ t_1 := x \cdot \left(x \cdot \left(-1 - t\_0\right)\right)\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{4 + \left(x \cdot \left(x \cdot \left(1 + t\_0\right)\right)\right) \cdot t\_1} \cdot \left(2 + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (* (* x x) (+ 0.08333333333333333 (* x (* x 0.002777777777777778)))))
        (t_1 (* x (* x (- -1.0 t_0)))))
   (if (<= x 6.5e+51)
     (* (/ 2.0 (+ 4.0 (* (* x (* x (+ 1.0 t_0))) t_1))) (+ 2.0 t_1))
     (/
      2.0
      (+ 2.0 (* x (* x (* 0.002777777777777778 (* x (* x (* x x)))))))))))

C

double code(double x) {
	double t_0 = (x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)));
	double t_1 = x * (x * (-1.0 - t_0));
	double tmp;
	if (x <= 6.5e+51) {
		tmp = (2.0 / (4.0 + ((x * (x * (1.0 + t_0))) * t_1))) * (2.0 + t_1);
	} else {
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
	}
	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) * (0.08333333333333333d0 + (x * (x * 0.002777777777777778d0)))
    t_1 = x * (x * ((-1.0d0) - t_0))
    if (x <= 6.5d+51) then
        tmp = (2.0d0 / (4.0d0 + ((x * (x * (1.0d0 + t_0))) * t_1))) * (2.0d0 + t_1)
    else
        tmp = 2.0d0 / (2.0d0 + (x * (x * (0.002777777777777778d0 * (x * (x * (x * x)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)));
	double t_1 = x * (x * (-1.0 - t_0));
	double tmp;
	if (x <= 6.5e+51) {
		tmp = (2.0 / (4.0 + ((x * (x * (1.0 + t_0))) * t_1))) * (2.0 + t_1);
	} else {
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)))
	t_1 = x * (x * (-1.0 - t_0))
	tmp = 0
	if x <= 6.5e+51:
		tmp = (2.0 / (4.0 + ((x * (x * (1.0 + t_0))) * t_1))) * (2.0 + t_1)
	else:
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(x * Float64(x * 0.002777777777777778))))
	t_1 = Float64(x * Float64(x * Float64(-1.0 - t_0)))
	tmp = 0.0
	if (x <= 6.5e+51)
		tmp = Float64(Float64(2.0 / Float64(4.0 + Float64(Float64(x * Float64(x * Float64(1.0 + t_0))) * t_1))) * Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * Float64(x * Float64(0.002777777777777778 * Float64(x * Float64(x * Float64(x * x))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)));
	t_1 = x * (x * (-1.0 - t_0));
	tmp = 0.0;
	if (x <= 6.5e+51)
		tmp = (2.0 / (4.0 + ((x * (x * (1.0 + t_0))) * t_1))) * (2.0 + t_1);
	else
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(x * N[(x * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.5e+51], N[(N[(2.0 / N[(4.0 + N[(N[(x * N[(x * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 + N[(x * N[(x * N[(0.002777777777777778 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 91.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 91.4%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{2}{\frac{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)}{\color{blue}{2 - \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)} \cdot \color{blue}{\left(2 - \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)}\right), \color{blue}{\left(2 - \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)}\right) \]

   7. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{\frac{2}{4 - \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right)} \cdot \left(2 - x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right)} \]

   if 6.5e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right), \color{blue}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right) + 2}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{5}\right)}\right), 2\right)\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot {x}^{\left(4 + 1\right)}\right)\right), 2\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left({x}^{4} \cdot x\right)\right)\right), 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{4}\right) \cdot x\right)\right), 2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot x\right)\right), 2\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot x\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \frac{1}{360}\right)\right)\right)\right), 2\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{4} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{4}\right), \frac{1}{360}\right)\right)\right), 2\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.002777777777777778\right)\right)} + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{4 + \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-1 - \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right)} \cdot \left(2 + x \cdot \left(x \cdot \left(-1 - \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{2 + \frac{\left(x \cdot x\right) \cdot \left(1 - x \cdot \left(t\_1 \cdot \left(x \cdot t\_0\right)\right)\right)}{1 - t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.08333333333333333 (* x (* x 0.002777777777777778))))
        (t_1 (* (* x x) t_0)))
   (if (<= x 5e+76)
     (/
      2.0
      (+ 2.0 (/ (* (* x x) (- 1.0 (* x (* t_1 (* x t_0))))) (- 1.0 t_1))))
     (/ 24.0 (* x (* x (* x x)))))))

C

double code(double x) {
	double t_0 = 0.08333333333333333 + (x * (x * 0.002777777777777778));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 5e+76) {
		tmp = 2.0 / (2.0 + (((x * x) * (1.0 - (x * (t_1 * (x * t_0))))) / (1.0 - t_1)));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	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 = 0.08333333333333333d0 + (x * (x * 0.002777777777777778d0))
    t_1 = (x * x) * t_0
    if (x <= 5d+76) then
        tmp = 2.0d0 / (2.0d0 + (((x * x) * (1.0d0 - (x * (t_1 * (x * t_0))))) / (1.0d0 - t_1)))
    else
        tmp = 24.0d0 / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.08333333333333333 + (x * (x * 0.002777777777777778));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 5e+76) {
		tmp = 2.0 / (2.0 + (((x * x) * (1.0 - (x * (t_1 * (x * t_0))))) / (1.0 - t_1)));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.08333333333333333 + (x * (x * 0.002777777777777778))
	t_1 = (x * x) * t_0
	tmp = 0
	if x <= 5e+76:
		tmp = 2.0 / (2.0 + (((x * x) * (1.0 - (x * (t_1 * (x * t_0))))) / (1.0 - t_1)))
	else:
		tmp = 24.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.08333333333333333 + Float64(x * Float64(x * 0.002777777777777778)))
	t_1 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= 5e+76)
		tmp = Float64(2.0 / Float64(2.0 + Float64(Float64(Float64(x * x) * Float64(1.0 - Float64(x * Float64(t_1 * Float64(x * t_0))))) / Float64(1.0 - t_1))));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.08333333333333333 + (x * (x * 0.002777777777777778));
	t_1 = (x * x) * t_0;
	tmp = 0.0;
	if (x <= 5e+76)
		tmp = 2.0 / (2.0 + (((x * x) * (1.0 - (x * (t_1 * (x * t_0))))) / (1.0 - t_1)));
	else
		tmp = 24.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(x * N[(x * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, 5e+76], N[(2.0 / N[(2.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 - N[(x * N[(t$95$1 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $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 < 4.99999999999999991e76

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 91.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(1 - x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 63.8%

      \[\leadsto \frac{2}{2 + \color{blue}{\frac{\left(1 - x \cdot \left(\left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{1 - \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)}}} \]

   if 4.99999999999999991e76 < 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2} \cdot 1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      7. pow-sqr N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{2} \cdot 1}{{x}^{2}} \cdot \frac{{x}^{2}}{1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. /-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-in N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

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

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

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   10. 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. unpow2 N/A

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

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot {x}^{\color{blue}{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 100.0%

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

   11. 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 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{2 + \frac{\left(x \cdot x\right) \cdot \left(1 - x \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right) \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{16 - t\_0 \cdot t\_0}{\left(4 + t\_0\right) \cdot \left(2 - x \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x 2.4e+51)
     (/ 2.0 (/ (- 16.0 (* t_0 t_0)) (* (+ 4.0 t_0) (- 2.0 (* x x)))))
     (/ 2.0 (+ 2.0 (* x (* x (* 0.002777777777777778 t_0))))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 2.4e+51) {
		tmp = 2.0 / ((16.0 - (t_0 * t_0)) / ((4.0 + t_0) * (2.0 - (x * x))));
	} else {
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= 2.4d+51) then
        tmp = 2.0d0 / ((16.0d0 - (t_0 * t_0)) / ((4.0d0 + t_0) * (2.0d0 - (x * x))))
    else
        tmp = 2.0d0 / (2.0d0 + (x * (x * (0.002777777777777778d0 * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 2.4e+51) {
		tmp = 2.0 / ((16.0 - (t_0 * t_0)) / ((4.0 + t_0) * (2.0 - (x * x))));
	} else {
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * t_0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= 2.4e+51:
		tmp = 2.0 / ((16.0 - (t_0 * t_0)) / ((4.0 + t_0) * (2.0 - (x * x))))
	else:
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * t_0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 2.4e+51)
		tmp = Float64(2.0 / Float64(Float64(16.0 - Float64(t_0 * t_0)) / Float64(Float64(4.0 + t_0) * Float64(2.0 - Float64(x * x)))));
	else
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * Float64(x * Float64(0.002777777777777778 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= 2.4e+51)
		tmp = 2.0 / ((16.0 - (t_0 * t_0)) / ((4.0 + t_0) * (2.0 - (x * x))));
	else
		tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.4e+51], N[(2.0 / N[(N[(16.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 + t$95$0), $MachinePrecision] * N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 + N[(x * N[(x * N[(0.002777777777777778 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e51

   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 77.7%

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

   5. Simplified 77.7%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]
   6. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(16 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 1}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}} \]

   if 2.3999999999999999e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right), \color{blue}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right) + 2}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{5}\right)}\right), 2\right)\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot {x}^{\left(4 + 1\right)}\right)\right), 2\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left({x}^{4} \cdot x\right)\right)\right), 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{4}\right) \cdot x\right)\right), 2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot x\right)\right), 2\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot x\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \frac{1}{360}\right)\right)\right)\right), 2\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{4} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{4}\right), \frac{1}{360}\right)\right)\right), 2\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.002777777777777778\right)\right)} + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{16 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (+
   2.0
   (*
    (* x x)
    (+
     1.0
     (* x (* x (+ 0.08333333333333333 (* x (* x 0.002777777777777778))))))))))

C

double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + (x * (x * 0.002777777777777778))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + (x * (x * (0.08333333333333333d0 + (x * (x * 0.002777777777777778d0))))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + (x * (x * 0.002777777777777778))))))));
}

Python

def code(x):
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + (x * (x * 0.002777777777777778))))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * 0.002777777777777778)))))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + (x * (x * 0.002777777777777778))))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.08333333333333333 + N[(x * N[(x * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.4%

   \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
6. Add Preprocessing

Alternative 6: 92.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (+ 2.0 (* (* x x) (+ 1.0 (* x (* x (* x (* x 0.002777777777777778)))))))))

C

double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (x * (x * 0.002777777777777778)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + (x * (x * (x * (x * 0.002777777777777778d0)))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (x * (x * 0.002777777777777778)))))));
}

Python

def code(x):
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (x * (x * 0.002777777777777778)))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(x * Float64(x * 0.002777777777777778))))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (x * (x * 0.002777777777777778)))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.4%

   \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 93.2%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 93.2%

   \[\leadsto \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)}\right)} \]
9. Add Preprocessing

Alternative 7: 69.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (+ 1.0 (* (* x x) (+ -0.5 (* (* x x) 0.20833333333333334))))
   (/ 2.0 (* x (* x (+ 1.0 (* x (* x 0.08333333333333333))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = 1.0d0 + ((x * x) * ((-0.5d0) + ((x * x) * 0.20833333333333334d0)))
    else
        tmp = 2.0d0 / (x * (x * (1.0d0 + (x * (x * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	} else {
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)))
	else:
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * 0.20833333333333334))));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = 1.0 + ((x * x) * (-0.5 + ((x * x) * 0.20833333333333334)));
	else
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(1.0 + N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   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 57.8%

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

   5. Simplified 57.8%

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

   if 1.44999999999999996 < 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 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2} \cdot 1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      7. pow-sqr N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{2} \cdot 1}{{x}^{2}} \cdot \frac{{x}^{2}}{1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. /-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-in N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

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

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

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

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

   8. Simplified 79.6%

      \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 79.6%

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

   10. Applied egg-rr 79.6%

       \[\leadsto \frac{2}{x \cdot \left(x \cdot \left(1 + \color{blue}{\left(x \cdot 0.08333333333333333\right) \cdot x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% 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 57.8%

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

   5. Simplified 57.8%

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

   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 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2} \cdot 1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      7. pow-sqr N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{2} \cdot 1}{{x}^{2}} \cdot \frac{{x}^{2}}{1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. /-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-in N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

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

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

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

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

   8. Simplified 79.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   10. 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. unpow2 N/A

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

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot {x}^{\color{blue}{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 79.6%

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

   11. Simplified 79.6%

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

Alternative 9: 91.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (+ 2.0 (* x (* x (* 0.002777777777777778 (* x (* x (* x x)))))))))

C

double code(double x) {
	return 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + (x * (x * (0.002777777777777778d0 * (x * (x * (x * x)))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * Float64(x * Float64(0.002777777777777778 * Float64(x * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + (x * (x * (0.002777777777777778 * (x * (x * (x * x)))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * N[(x * N[(0.002777777777777778 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.4%

   \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right) + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right), \color{blue}{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{12} + x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   12. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{360}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

7. Applied egg-rr 93.4%

   \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)\right) + 2}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{5}\right)}\right), 2\right)\right) \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot {x}^{\left(4 + 1\right)}\right)\right), 2\right)\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{360} \cdot \left({x}^{4} \cdot x\right)\right)\right), 2\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{4}\right) \cdot x\right)\right), 2\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot x\right)\right), 2\right)\right) \]

   5. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{360} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot x\right)\right), 2\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)\right), 2\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right), 2\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \frac{1}{360}\right)\right)\right)\right), 2\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

   12. pow-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{4} \cdot \frac{1}{360}\right)\right)\right), 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{4}\right), \frac{1}{360}\right)\right)\right), 2\right)\right) \]

10. Simplified 93.1%

    \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.002777777777777778\right)\right)} + 2} \]

11. Final simplification 93.1%

    \[\leadsto \frac{2}{2 + x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 10: 88.3% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 88.1%

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

7. Simplified 88.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 88.1%

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

9. Applied egg-rr 88.1%

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

10. Final simplification 88.1%

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

Alternative 11: 82.4% 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 80.1%

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

   5. Simplified 80.1%

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

   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 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2} \cdot 1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      7. pow-sqr N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{x}^{2} \cdot 1}{{x}^{2}} \cdot \frac{{x}^{2}}{1} + {x}^{4} \cdot \frac{1}{12}\right)\right) \]

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. /-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-in N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

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

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

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

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

   8. Simplified 79.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   10. 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. unpow2 N/A

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

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot {x}^{\color{blue}{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 79.6%

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

   11. Simplified 79.6%

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

Alternative 12: 63.9% 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 58.3%

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

   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 60.1%

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

   5. Simplified 60.1%

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

   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 60.1%

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

   8. Simplified 60.1%

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

Alternative 13: 76.4% 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 75.1%

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

5. Simplified 75.1%

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

Alternative 14: 51.2% 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 44.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))