Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 17

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. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 74.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot t\_1\\ t_3 := x \cdot t\_1\\ \mathbf{if}\;x \leq 6.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - t\_2 \cdot t\_2\right) \cdot \left(16 - t\_0 \cdot t\_0\right)}{\left(8 - t\_2\right) \cdot \left(t\_0 + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + \left(x \cdot x\right) \cdot t\_3\right) \cdot \left(4 - t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (+ (* x x) -2.0)))
        (t_1 (* x (* x x)))
        (t_2 (* t_1 t_1))
        (t_3 (* x t_1)))
   (if (<= x 6.6e+30)
     (/
      32.0
      (/
       (* (- 64.0 (* t_2 t_2)) (- 16.0 (* t_0 t_0)))
       (* (- 8.0 t_2) (+ t_0 4.0))))
     (/ 32.0 (* (+ 8.0 (* (* x x) t_3)) (- 4.0 t_3))))))

C

double code(double x) {
	double t_0 = (x * x) * ((x * x) + -2.0);
	double t_1 = x * (x * x);
	double t_2 = t_1 * t_1;
	double t_3 = x * t_1;
	double tmp;
	if (x <= 6.6e+30) {
		tmp = 32.0 / (((64.0 - (t_2 * t_2)) * (16.0 - (t_0 * t_0))) / ((8.0 - t_2) * (t_0 + 4.0)));
	} else {
		tmp = 32.0 / ((8.0 + ((x * x) * t_3)) * (4.0 - t_3));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (x * x) * ((x * x) + (-2.0d0))
    t_1 = x * (x * x)
    t_2 = t_1 * t_1
    t_3 = x * t_1
    if (x <= 6.6d+30) then
        tmp = 32.0d0 / (((64.0d0 - (t_2 * t_2)) * (16.0d0 - (t_0 * t_0))) / ((8.0d0 - t_2) * (t_0 + 4.0d0)))
    else
        tmp = 32.0d0 / ((8.0d0 + ((x * x) * t_3)) * (4.0d0 - t_3))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * ((x * x) + -2.0);
	double t_1 = x * (x * x);
	double t_2 = t_1 * t_1;
	double t_3 = x * t_1;
	double tmp;
	if (x <= 6.6e+30) {
		tmp = 32.0 / (((64.0 - (t_2 * t_2)) * (16.0 - (t_0 * t_0))) / ((8.0 - t_2) * (t_0 + 4.0)));
	} else {
		tmp = 32.0 / ((8.0 + ((x * x) * t_3)) * (4.0 - t_3));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * ((x * x) + -2.0)
	t_1 = x * (x * x)
	t_2 = t_1 * t_1
	t_3 = x * t_1
	tmp = 0
	if x <= 6.6e+30:
		tmp = 32.0 / (((64.0 - (t_2 * t_2)) * (16.0 - (t_0 * t_0))) / ((8.0 - t_2) * (t_0 + 4.0)))
	else:
		tmp = 32.0 / ((8.0 + ((x * x) * t_3)) * (4.0 - t_3))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(Float64(x * x) + -2.0))
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(x * t_1)
	tmp = 0.0
	if (x <= 6.6e+30)
		tmp = Float64(32.0 / Float64(Float64(Float64(64.0 - Float64(t_2 * t_2)) * Float64(16.0 - Float64(t_0 * t_0))) / Float64(Float64(8.0 - t_2) * Float64(t_0 + 4.0))));
	else
		tmp = Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * t_3)) * Float64(4.0 - t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * ((x * x) + -2.0);
	t_1 = x * (x * x);
	t_2 = t_1 * t_1;
	t_3 = x * t_1;
	tmp = 0.0;
	if (x <= 6.6e+30)
		tmp = 32.0 / (((64.0 - (t_2 * t_2)) * (16.0 - (t_0 * t_0))) / ((8.0 - t_2) * (t_0 + 4.0)));
	else
		tmp = 32.0 / ((8.0 + ((x * x) * t_3)) * (4.0 - t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[x, 6.6e+30], N[(32.0 / N[(N[(N[(64.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(16.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(8.0 - t$95$2), $MachinePrecision] * N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(4.0 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.60000000000000053e30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 81.9%

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

   5. Simplified 81.9%

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

      1. flip3-+ N/A

         \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

      4. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 92.2%

       \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(\color{blue}{4} - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right) \]

       2. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot \frac{4 \cdot 4 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}{\color{blue}{4 + \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)}}\right)\right) \]

       3. frac-times N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 \cdot 4 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)}{\color{blue}{\left(8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{/.f64}\left(\left(\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 \cdot 4 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right), \color{blue}{\left(\left(8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 70.8%

       \[\leadsto \frac{32}{\color{blue}{\frac{\left(64 - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(16 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)}{\left(8 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}}} \]

   if 6.60000000000000053e30 < 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 45.4%

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

   5. Simplified 45.4%

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

      1. flip3-+ N/A

         \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

      4. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

   7. Applied egg-rr 4.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \color{blue}{\left({x}^{4}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(16 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)}{\left(8 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right) + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - t\_1 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(t\_0 \cdot t\_0 + 8\right)}{16 + t\_1 \cdot \left(t\_1 + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot t\_0\right)\right) \cdot \left(4 - x \cdot \left(x \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* (* x x) (+ (* x x) -2.0))))
   (if (<= x 4e+38)
     (/
      32.0
      (/
       (* (- 64.0 (* t_1 (* t_1 t_1))) (+ (* t_0 t_0) 8.0))
       (+ 16.0 (* t_1 (+ t_1 4.0)))))
     (/ 32.0 (* (+ 8.0 (* (* x x) (* x t_0))) (- 4.0 (* x (* x -2.0))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * ((x * x) + -2.0);
	double tmp;
	if (x <= 4e+38) {
		tmp = 32.0 / (((64.0 - (t_1 * (t_1 * t_1))) * ((t_0 * t_0) + 8.0)) / (16.0 + (t_1 * (t_1 + 4.0))));
	} else {
		tmp = 32.0 / ((8.0 + ((x * x) * (x * t_0))) * (4.0 - (x * (x * -2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * x) * ((x * x) + (-2.0d0))
    if (x <= 4d+38) then
        tmp = 32.0d0 / (((64.0d0 - (t_1 * (t_1 * t_1))) * ((t_0 * t_0) + 8.0d0)) / (16.0d0 + (t_1 * (t_1 + 4.0d0))))
    else
        tmp = 32.0d0 / ((8.0d0 + ((x * x) * (x * t_0))) * (4.0d0 - (x * (x * (-2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * ((x * x) + -2.0);
	double tmp;
	if (x <= 4e+38) {
		tmp = 32.0 / (((64.0 - (t_1 * (t_1 * t_1))) * ((t_0 * t_0) + 8.0)) / (16.0 + (t_1 * (t_1 + 4.0))));
	} else {
		tmp = 32.0 / ((8.0 + ((x * x) * (x * t_0))) * (4.0 - (x * (x * -2.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) * ((x * x) + -2.0)
	tmp = 0
	if x <= 4e+38:
		tmp = 32.0 / (((64.0 - (t_1 * (t_1 * t_1))) * ((t_0 * t_0) + 8.0)) / (16.0 + (t_1 * (t_1 + 4.0))))
	else:
		tmp = 32.0 / ((8.0 + ((x * x) * (x * t_0))) * (4.0 - (x * (x * -2.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * Float64(Float64(x * x) + -2.0))
	tmp = 0.0
	if (x <= 4e+38)
		tmp = Float64(32.0 / Float64(Float64(Float64(64.0 - Float64(t_1 * Float64(t_1 * t_1))) * Float64(Float64(t_0 * t_0) + 8.0)) / Float64(16.0 + Float64(t_1 * Float64(t_1 + 4.0)))));
	else
		tmp = Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * Float64(x * t_0))) * Float64(4.0 - Float64(x * Float64(x * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) * ((x * x) + -2.0);
	tmp = 0.0;
	if (x <= 4e+38)
		tmp = 32.0 / (((64.0 - (t_1 * (t_1 * t_1))) * ((t_0 * t_0) + 8.0)) / (16.0 + (t_1 * (t_1 + 4.0))));
	else
		tmp = 32.0 / ((8.0 + ((x * x) * (x * t_0))) * (4.0 - (x * (x * -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+38], N[(32.0 / N[(N[(N[(64.0 - N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 + N[(t$95$1 * N[(t$95$1 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999991e38

   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 80.7%

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

   5. Simplified 80.7%

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

      1. flip3-+ N/A

         \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

      4. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

   7. Applied egg-rr 67.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 92.3%

       \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \color{blue}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right)\right) \]

       2. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{{4}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}^{3}}{4 \cdot 4 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) + 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)} \cdot \left(\color{blue}{8} + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{\left({4}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}^{3}\right) \cdot \left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\color{blue}{4 \cdot 4 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) + 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{/.f64}\left(\left(\left({4}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}^{3}\right) \cdot \left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \color{blue}{\left(4 \cdot 4 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) + 4 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 71.2%

       \[\leadsto \frac{32}{\color{blue}{\frac{\left(64 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right) \cdot \left(8 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{16 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}}} \]

   if 3.99999999999999991e38 < 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 47.6%

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

   5. Simplified 47.6%

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

      1. flip3-+ N/A

         \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

      4. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \color{blue}{\left(-2 \cdot {x}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{2} \cdot \color{blue}{-2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(\left(x \cdot x\right) \cdot -2\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot -2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot -2\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \color{blue}{x \cdot \left(x \cdot -2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + 8\right)}{16 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x + -2\right) + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot -2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - t\_1 \cdot t\_1\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}{8 - t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* t_0 t_0)))
   (if (<= x 2.4e+51)
     (/
      32.0
      (/
       (* (- 64.0 (* t_1 t_1)) (- 4.0 (* (* x x) (+ (* x x) -2.0))))
       (- 8.0 t_1)))
     (/
      2.0
      (+ 2.0 (* x (* (* x x) (* x (* (* x x) 0.002777777777777778)))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = t_0 * t_0;
	double tmp;
	if (x <= 2.4e+51) {
		tmp = 32.0 / (((64.0 - (t_1 * t_1)) * (4.0 - ((x * x) * ((x * x) + -2.0)))) / (8.0 - t_1));
	} else {
		tmp = 2.0 / (2.0 + (x * ((x * x) * (x * ((x * x) * 0.002777777777777778)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = t_0 * t_0
    if (x <= 2.4d+51) then
        tmp = 32.0d0 / (((64.0d0 - (t_1 * t_1)) * (4.0d0 - ((x * x) * ((x * x) + (-2.0d0))))) / (8.0d0 - t_1))
    else
        tmp = 2.0d0 / (2.0d0 + (x * ((x * x) * (x * ((x * x) * 0.002777777777777778d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = t_0 * t_0;
	double tmp;
	if (x <= 2.4e+51) {
		tmp = 32.0 / (((64.0 - (t_1 * t_1)) * (4.0 - ((x * x) * ((x * x) + -2.0)))) / (8.0 - t_1));
	} else {
		tmp = 2.0 / (2.0 + (x * ((x * x) * (x * ((x * x) * 0.002777777777777778)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = t_0 * t_0
	tmp = 0
	if x <= 2.4e+51:
		tmp = 32.0 / (((64.0 - (t_1 * t_1)) * (4.0 - ((x * x) * ((x * x) + -2.0)))) / (8.0 - t_1))
	else:
		tmp = 2.0 / (2.0 + (x * ((x * x) * (x * ((x * x) * 0.002777777777777778)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (x <= 2.4e+51)
		tmp = Float64(32.0 / Float64(Float64(Float64(64.0 - Float64(t_1 * t_1)) * Float64(4.0 - Float64(Float64(x * x) * Float64(Float64(x * x) + -2.0)))) / Float64(8.0 - t_1)));
	else
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * x) * Float64(x * Float64(Float64(x * x) * 0.002777777777777778))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = t_0 * t_0;
	tmp = 0.0;
	if (x <= 2.4e+51)
		tmp = 32.0 / (((64.0 - (t_1 * t_1)) * (4.0 - ((x * x) * ((x * x) + -2.0)))) / (8.0 - t_1));
	else
		tmp = 2.0 / (2.0 + (x * ((x * x) * (x * ((x * x) * 0.002777777777777778)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[x, 2.4e+51], N[(32.0 / N[(N[(N[(64.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(8.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $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 79.6%

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

   5. Simplified 79.6%

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

      1. flip3-+ N/A

         \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

      4. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

   7. Applied egg-rr 66.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 92.5%

       \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(\color{blue}{4} - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(32, \left(\frac{\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}{\color{blue}{8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(32, \mathsf{/.f64}\left(\left(\left(8 \cdot 8 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)\right), \color{blue}{\left(8 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 72.7%

       \[\leadsto \frac{32}{\color{blue}{\frac{\left(64 - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)}{8 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}}} \]

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. unpow3 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 5: 95.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  32.0
  (*
   (+ 8.0 (* (* x x) (* x (* x (* x x)))))
   (- 4.0 (* (* x x) (+ (* x x) -2.0))))))

C

double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - ((x * x) * ((x * x) + -2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0 / ((8.0d0 + ((x * x) * (x * (x * (x * x))))) * (4.0d0 - ((x * x) * ((x * x) + (-2.0d0)))))
end function

Java

public static double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - ((x * x) * ((x * x) + -2.0))));
}

Python

def code(x):
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - ((x * x) * ((x * x) + -2.0))))

Julia

function code(x)
	return Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) * Float64(4.0 - Float64(Float64(x * x) * Float64(Float64(x * x) + -2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - ((x * x) * ((x * x) + -2.0))));
end

Wolfram

code[x_] := N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -2.0), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
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 73.1%

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

5. Simplified 73.1%

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

   1. flip3-+ N/A

      \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

   4. frac-times N/A

      \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

7. Applied egg-rr 51.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.1%

    \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]
11. Add Preprocessing

Alternative 6: 94.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \frac{32}{\left(8 + \left(x \cdot x\right) \cdot t\_0\right) \cdot \left(4 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (/ 32.0 (* (+ 8.0 (* (* x x) t_0)) (- 4.0 t_0)))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	return 32.0 / ((8.0 + ((x * x) * t_0)) * (4.0 - t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * (x * x))
    code = 32.0d0 / ((8.0d0 + ((x * x) * t_0)) * (4.0d0 - t_0))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	return 32.0 / ((8.0 + ((x * x) * t_0)) * (4.0 - t_0));
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	return 32.0 / ((8.0 + ((x * x) * t_0)) * (4.0 - t_0))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	return Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * t_0)) * Float64(4.0 - t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (x * x));
	tmp = 32.0 / ((8.0 + ((x * x) * t_0)) * (4.0 - t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(4.0 - t$95$0), $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 73.1%

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

5. Simplified 73.1%

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

   1. flip3-+ N/A

      \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

   4. frac-times N/A

      \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

7. Applied egg-rr 51.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.1%

    \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \color{blue}{\left({x}^{4}\right)}\right)\right)\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

    6. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

    8. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 93.6%

        \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

13. Simplified 93.6%

    \[\leadsto \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)} \]
14. Add Preprocessing

Alternative 7: 94.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot -2\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 32.0 (* (+ 8.0 (* (* x x) (* x (* x (* x x))))) (- 4.0 (* x (* x -2.0))))))

C

double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * -2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0 / ((8.0d0 + ((x * x) * (x * (x * (x * x))))) * (4.0d0 - (x * (x * (-2.0d0)))))
end function

Java

public static double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * -2.0))));
}

Python

def code(x):
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * -2.0))))

Julia

function code(x)
	return Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) * Float64(4.0 - Float64(x * Float64(x * -2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * -2.0))));
end

Wolfram

code[x_] := N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(x * N[(x * -2.0), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
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 73.1%

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

5. Simplified 73.1%

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

   1. flip3-+ N/A

      \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

   4. frac-times N/A

      \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

7. Applied egg-rr 51.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.1%

    \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \color{blue}{\left(-2 \cdot {x}^{2}\right)}\right)\right)\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left({x}^{2} \cdot \color{blue}{-2}\right)\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(\left(x \cdot x\right) \cdot -2\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot -2\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot -2\right)}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 92.7%

       \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{-2}\right)\right)\right)\right)\right) \]

13. Simplified 92.7%

    \[\leadsto \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \color{blue}{x \cdot \left(x \cdot -2\right)}\right)} \]
14. Add Preprocessing

Alternative 8: 92.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 90.8%

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

7. Simplified 90.8%

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

Alternative 9: 92.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.001388888888888889), $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. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 90.8%

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

7. Simplified 90.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   10. unpow3 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 90.4%

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

10. Simplified 90.4%

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

11. Final simplification 90.4%

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

Alternative 10: 69.8% accurate, 11.4× speedup

Math

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

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(x * Float64(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[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 68.1%

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

   5. Simplified 68.1%

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

   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 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.3%

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

   8. Simplified 77.3%

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

Alternative 11: 91.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $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. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 90.8%

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

5. Simplified 90.8%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. unpow3 N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. unpow3 N/A

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

   16. unpow2 N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

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

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

   20. *-commutative N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 89.9%

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

8. Simplified 89.9%

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

Alternative 12: 88.1% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{32}{8 \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 32.0 (* 8.0 (- 4.0 (* (* x x) (+ (* x x) -2.0))))))

C

double code(double x) {
	return 32.0 / (8.0 * (4.0 - ((x * x) * ((x * x) + -2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0 / (8.0d0 * (4.0d0 - ((x * x) * ((x * x) + (-2.0d0)))))
end function

Java

public static double code(double x) {
	return 32.0 / (8.0 * (4.0 - ((x * x) * ((x * x) + -2.0))));
}

Python

def code(x):
	return 32.0 / (8.0 * (4.0 - ((x * x) * ((x * x) + -2.0))))

Julia

function code(x)
	return Float64(32.0 / Float64(8.0 * Float64(4.0 - Float64(Float64(x * x) * Float64(Float64(x * x) + -2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 32.0 / (8.0 * (4.0 - ((x * x) * ((x * x) + -2.0))));
end

Wolfram

code[x_] := N[(32.0 / N[(8.0 * N[(4.0 - N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -2.0), $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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
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 73.1%

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

5. Simplified 73.1%

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

   1. flip3-+ N/A

      \[\leadsto \frac{2}{\frac{{2}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(2 \cdot 2 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \frac{2}{{2}^{3} + {\left(x \cdot x\right)}^{3}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}{\color{blue}{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)}} \]

   4. frac-times N/A

      \[\leadsto \frac{2 \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left({2}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 2 \cdot \left(x \cdot x\right)\right)\right)}} \]

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

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

7. Applied egg-rr 51.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.1%

    \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(32, \mathsf{*.f64}\left(\color{blue}{8}, \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

13. Simplified 87.1%

    \[\leadsto \frac{32}{\color{blue}{8} \cdot \left(4 - \left(x \cdot x\right) \cdot \left(x \cdot x + -2\right)\right)} \]
14. Add Preprocessing

Alternative 13: 88.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
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 86.7%

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

5. Simplified 86.7%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 86.7%

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

7. Applied egg-rr 86.7%

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

8. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 86.7%

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

10. Simplified 86.7%

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

Alternative 14: 82.3% accurate, 14.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = 2.0 / ((x * x) + 2.0);
	} 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 / ((x * x) + 2.0d0)
    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 / ((x * x) + 2.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.7:
		tmp = 2.0 / ((x * x) + 2.0)
	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(Float64(x * x) + 2.0));
	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 / ((x * x) + 2.0);
	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[(N[(x * x), $MachinePrecision] + 2.0), $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 83.6%

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

   5. Simplified 83.6%

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

   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 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.3%

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

   8. Simplified 77.3%

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

4. Final simplification 82.0%

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

Alternative 15: 64.2% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.42) 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		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 <= 1.42d0) 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 <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		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 <= 1.42)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], 1.0, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.8%

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

   if 1.4199999999999999 < 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 42.9%

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

   5. Simplified 42.9%

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

   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 42.9%

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

   8. Simplified 42.9%

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

Alternative 16: 76.8% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(x * x), $MachinePrecision] + 2.0), $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 73.1%

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

5. Simplified 73.1%

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

6. Final simplification 73.1%

   \[\leadsto \frac{2}{x \cdot x + 2} \]
7. Add Preprocessing

Alternative 17: 51.9% 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 51.1%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))