Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 16

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: 75.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{16}{\frac{\left(64 - t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + 2\right)\right)}{16 + t\_0 \cdot \left(t\_0 + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x 3.4e+38)
     (/
      16.0
      (/
       (* (- 64.0 (* t_0 (* t_0 t_0))) (+ 4.0 (* (* x x) (+ (* x x) 2.0))))
       (+ 16.0 (* t_0 (+ t_0 4.0)))))
     (/ -16.0 (* (* x x) (* (* x x) t_0))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 3.4e+38) {
		tmp = 16.0 / (((64.0 - (t_0 * (t_0 * t_0))) * (4.0 + ((x * x) * ((x * x) + 2.0)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= 3.4d+38) then
        tmp = 16.0d0 / (((64.0d0 - (t_0 * (t_0 * t_0))) * (4.0d0 + ((x * x) * ((x * x) + 2.0d0)))) / (16.0d0 + (t_0 * (t_0 + 4.0d0))))
    else
        tmp = (-16.0d0) / ((x * x) * ((x * x) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 3.4e+38) {
		tmp = 16.0 / (((64.0 - (t_0 * (t_0 * t_0))) * (4.0 + ((x * x) * ((x * x) + 2.0)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= 3.4e+38:
		tmp = 16.0 / (((64.0 - (t_0 * (t_0 * t_0))) * (4.0 + ((x * x) * ((x * x) + 2.0)))) / (16.0 + (t_0 * (t_0 + 4.0))))
	else:
		tmp = -16.0 / ((x * x) * ((x * x) * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 3.4e+38)
		tmp = Float64(16.0 / Float64(Float64(Float64(64.0 - Float64(t_0 * Float64(t_0 * t_0))) * Float64(4.0 + Float64(Float64(x * x) * Float64(Float64(x * x) + 2.0)))) / Float64(16.0 + Float64(t_0 * Float64(t_0 + 4.0)))));
	else
		tmp = Float64(-16.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= 3.4e+38)
		tmp = 16.0 / (((64.0 - (t_0 * (t_0 * t_0))) * (4.0 + ((x * x) * ((x * x) + 2.0)))) / (16.0 + (t_0 * (t_0 + 4.0))));
	else
		tmp = -16.0 / ((x * x) * ((x * x) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.4e+38], N[(16.0 / N[(N[(N[(64.0 - N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(16.0 + N[(t$95$0 * N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-16.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999996e38

   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 79.2%

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

   5. Simplified 79.2%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip3-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 92.7%

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

       1. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(16, \left(\frac{{4}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}}{4 \cdot 4 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 4 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \cdot \left(\color{blue}{4} + \left(x \cdot x\right) \cdot \left(2 + x \cdot x\right)\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(16, \left(\frac{\left({4}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(2 + x \cdot x\right)\right)}{\color{blue}{4 \cdot 4 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 4 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(16, \mathsf{/.f64}\left(\left(\left({4}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(2 + x \cdot x\right)\right)\right), \color{blue}{\left(4 \cdot 4 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 4 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 65.3%

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

   if 3.39999999999999996e38 < 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 69.8%

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

   5. Simplified 69.8%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip3-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 1.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-16}{{x}^{8}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-16, \color{blue}{\left({x}^{8}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{4} \cdot \color{blue}{{x}^{4}}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{\left(3 + 1\right)} \cdot {x}^{4}\right)\right) \]

       5. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left({x}^{3} \cdot x\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       8. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\color{blue}{{x}^{3}} \cdot {x}^{3}\right)\right)\right) \]

       12. cube-prod N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left({x}^{2}\right)}^{3}\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       15. pow-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{4} \cdot {x}^{2}\right)\right)\right) \]

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

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

       18. unpow2 N/A

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

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

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

       20. *-commutative N/A

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

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

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

       22. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right)\right) \]

       24. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

       25. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right) \]

       26. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

       28. cube-mult N/A

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

       29. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

       31. unpow2 N/A

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

       32. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{16}{\frac{\left(64 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + 2\right)\right)}{16 + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right) + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{if}\;x \leq 2.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)} \cdot \left(1 - x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           0.5
           (*
            (* x x)
            (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (if (<= x 2.5e+51)
     (* (/ 1.0 (- 1.0 (* (* x x) (* t_0 t_0)))) (- 1.0 (* x t_0)))
     (/ 720.0 (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	double tmp;
	if (x <= 2.5e+51) {
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))
    if (x <= 2.5d+51) then
        tmp = (1.0d0 / (1.0d0 - ((x * x) * (t_0 * t_0)))) * (1.0d0 - (x * t_0))
    else
        tmp = 720.0d0 / ((x * x) * (x * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	double tmp;
	if (x <= 2.5e+51) {
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))
	tmp = 0
	if x <= 2.5e+51:
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0))
	else:
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))
	tmp = 0.0
	if (x <= 2.5e+51)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) * Float64(1.0 - Float64(x * t_0)));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	tmp = 0.0;
	if (x <= 2.5e+51)
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
	else
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.5e+51], N[(N[(1.0 / N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e51

   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. Step-by-step derivation

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 66.0%

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

   if 2.5e51 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot {x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

Alternative 4: 77.1% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           0.5
           (*
            (* x x)
            (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (if (<= x 1e+77)
     (*
      (/ 1.0 (- 1.0 (* (* x x) (* t_0 t_0))))
      (- 1.0 (* x (* x (+ 0.5 (* x (* x 0.041666666666666664)))))))
     (/ 24.0 (* x (* x (* x x)))))))

C

double code(double x) {
	double t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	double tmp;
	if (x <= 1e+77) {
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * (x * (0.5 + (x * (x * 0.041666666666666664))))));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))
    if (x <= 1d+77) then
        tmp = (1.0d0 / (1.0d0 - ((x * x) * (t_0 * t_0)))) * (1.0d0 - (x * (x * (0.5d0 + (x * (x * 0.041666666666666664d0))))))
    else
        tmp = 24.0d0 / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	double tmp;
	if (x <= 1e+77) {
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * (x * (0.5 + (x * (x * 0.041666666666666664))))));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))
	tmp = 0
	if x <= 1e+77:
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * (x * (0.5 + (x * (x * 0.041666666666666664))))))
	else:
		tmp = 24.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))
	tmp = 0.0
	if (x <= 1e+77)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) * Float64(1.0 - Float64(x * Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))))));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))));
	tmp = 0.0;
	if (x <= 1e+77)
		tmp = (1.0 / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * (x * (0.5 + (x * (x * 0.041666666666666664))))));
	else
		tmp = 24.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+77], N[(N[(1.0 / N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(x * N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $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 < 9.99999999999999983e76

   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 91.1%

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

   7. Simplified 91.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 63.8%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. *-lowering-*.f64 69.4%

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

   12. Simplified 69.4%

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

   if 9.99999999999999983e76 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   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 100.0%

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

   8. Simplified 100.0%

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

Alternative 5: 92.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{1}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0)
   (/
    1.0
    (+
     1.0
     (*
      (* x x)
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (/ -16.0 (* (* x x) (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 1.0 / (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 1.0d0 / (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))))
    else
        tmp = (-16.0d0) / ((x * x) * ((x * x) * (x * (x * (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 1.0 / (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 700.0:
		tmp = 1.0 / (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))))
	else:
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 700.0)
		tmp = 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)))))));
	else
		tmp = Float64(-16.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 700.0)
		tmp = 1.0 / (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	else
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 700.0], 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], N[(-16.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 700

   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 94.3%

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

   7. Simplified 94.3%

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

   if 700 < 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 62.1%

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

   5. Simplified 62.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip3-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 2.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 88.9%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-16}{{x}^{8}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-16, \color{blue}{\left({x}^{8}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{4} \cdot \color{blue}{{x}^{4}}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{\left(3 + 1\right)} \cdot {x}^{4}\right)\right) \]

       5. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left({x}^{3} \cdot x\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       8. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\color{blue}{{x}^{3}} \cdot {x}^{3}\right)\right)\right) \]

       12. cube-prod N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left({x}^{2}\right)}^{3}\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       15. pow-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{4} \cdot {x}^{2}\right)\right)\right) \]

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

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

       18. unpow2 N/A

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

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

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

       20. *-commutative N/A

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

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

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

       22. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right)\right) \]

       24. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

       25. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right) \]

       26. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

       28. cube-mult N/A

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

       29. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

       31. unpow2 N/A

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

       32. *-lowering-*.f64 88.9%

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 88.9%

       \[\leadsto \color{blue}{\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{16}{\left(4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + 2\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 16.0 (* (- 4.0 (* x (* x (* x x)))) (+ 4.0 (* (* x x) (+ (* x x) 2.0))))))

C

double code(double x) {
	return 16.0 / ((4.0 - (x * (x * (x * x)))) * (4.0 + ((x * x) * ((x * x) + 2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 16.0d0 / ((4.0d0 - (x * (x * (x * x)))) * (4.0d0 + ((x * x) * ((x * x) + 2.0d0))))
end function

Java

public static double code(double x) {
	return 16.0 / ((4.0 - (x * (x * (x * x)))) * (4.0 + ((x * x) * ((x * x) + 2.0))));
}

Python

def code(x):
	return 16.0 / ((4.0 - (x * (x * (x * x)))) * (4.0 + ((x * x) * ((x * x) + 2.0))))

Julia

function code(x)
	return Float64(16.0 / Float64(Float64(4.0 - 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 = 16.0 / ((4.0 - (x * (x * (x * x)))) * (4.0 + ((x * x) * ((x * x) + 2.0))));
end

Wolfram

code[x_] := N[(16.0 / N[(N[(4.0 - N[(x * N[(x * N[(x * x), $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 77.3%

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

5. Simplified 77.3%

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

   1. flip-+ N/A

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

   2. associate-/r/ N/A

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

   3. flip3-- N/A

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

   4. frac-times N/A

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

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

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

7. Applied egg-rr 50.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.2%

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

11. Final simplification 94.2%

    \[\leadsto \frac{16}{\left(4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(4 + \left(x \cdot x\right) \cdot \left(x \cdot x + 2\right)\right)} \]
12. Add Preprocessing

Alternative 7: 91.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x + \left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0)
   (/ 2.0 (+ 2.0 (* x (+ x (* (* x x) (* x 0.08333333333333333))))))
   (/ -16.0 (* (* x x) (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 2.0d0 / (2.0d0 + (x * (x + ((x * x) * (x * 0.08333333333333333d0)))))
    else
        tmp = (-16.0d0) / ((x * x) * ((x * x) * (x * (x * (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	} else {
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 700.0:
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))))
	else:
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 700.0)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * Float64(x + Float64(Float64(x * x) * Float64(x * 0.08333333333333333))))));
	else
		tmp = Float64(-16.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 700.0)
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	else
		tmp = -16.0 / ((x * x) * ((x * x) * (x * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 700.0], N[(2.0 / N[(2.0 + N[(x * N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-16.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 700

   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 91.8%

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

   5. Simplified 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}} \]
   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 91.8%

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

   7. Applied egg-rr 91.8%

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

   if 700 < 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 62.1%

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

   5. Simplified 62.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip3-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 2.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 88.9%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-16}{{x}^{8}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-16, \color{blue}{\left({x}^{8}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{4} \cdot \color{blue}{{x}^{4}}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{\left(3 + 1\right)} \cdot {x}^{4}\right)\right) \]

       5. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left({x}^{3} \cdot x\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {\color{blue}{x}}^{4}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       8. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot {x}^{3}\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\color{blue}{{x}^{3}} \cdot {x}^{3}\right)\right)\right) \]

       12. cube-prod N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot {\left({x}^{2}\right)}^{3}\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       15. pow-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \left({x}^{2} \cdot \left({x}^{4} \cdot {x}^{2}\right)\right)\right) \]

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

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

       18. unpow2 N/A

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

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

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

       20. *-commutative N/A

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

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

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

       22. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right)\right) \]

       24. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

       25. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right) \]

       26. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

       28. cube-mult N/A

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

       29. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

       31. unpow2 N/A

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

       32. *-lowering-*.f64 88.9%

           \[\leadsto \mathsf{/.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 88.9%

       \[\leadsto \color{blue}{\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x + \left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \frac{16}{\left(t\_0 + 4\right) \cdot \left(4 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x))))) (/ 16.0 (* (+ t_0 4.0) (- 4.0 t_0)))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	return 16.0 / ((t_0 + 4.0) * (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 = 16.0d0 / ((t_0 + 4.0d0) * (4.0d0 - t_0))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	return 16.0 / ((t_0 + 4.0) * (4.0 - t_0));
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	return 16.0 / ((t_0 + 4.0) * (4.0 - t_0))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	return Float64(16.0 / Float64(Float64(t_0 + 4.0) * Float64(4.0 - t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (x * x));
	tmp = 16.0 / ((t_0 + 4.0) * (4.0 - t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(16.0 / N[(N[(t$95$0 + 4.0), $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 77.3%

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

5. Simplified 77.3%

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

   1. flip-+ N/A

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

   2. associate-/r/ N/A

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

   3. flip3-- N/A

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

   4. frac-times N/A

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

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

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

7. Applied egg-rr 50.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{16}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 94.2%

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

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \color{blue}{\left({x}^{4}\right)}\right)\right)\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

    2. pow-plus N/A

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

    5. cube-mult N/A

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

    6. unpow2 N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

    8. unpow2 N/A

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

    9. *-lowering-*.f64 94.0%

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

13. Simplified 94.0%

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

14. Final simplification 94.0%

    \[\leadsto \frac{16}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right) + 4\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
15. Add Preprocessing

Alternative 9: 90.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x + \left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.2)
   (/ 2.0 (+ 2.0 (* x (+ x (* (* x x) (* x 0.08333333333333333))))))
   (/ 720.0 (* (* x x) (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 6.2) {
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.2d0) then
        tmp = 2.0d0 / (2.0d0 + (x * (x + ((x * x) * (x * 0.08333333333333333d0)))))
    else
        tmp = 720.0d0 / ((x * x) * (x * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.2) {
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.2:
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))))
	else:
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.2)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * Float64(x + Float64(Float64(x * x) * Float64(x * 0.08333333333333333))))));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.2)
		tmp = 2.0 / (2.0 + (x * (x + ((x * x) * (x * 0.08333333333333333)))));
	else
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.2], N[(2.0 / N[(2.0 + N[(x * N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000018

   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 92.2%

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

   5. Simplified 92.2%

      \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}} \]
   6. 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 92.2%

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

   7. Applied egg-rr 92.2%

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

   if 6.20000000000000018 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. 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 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot {x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 86.0%

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

   10. Simplified 86.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x + \left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (+ 1.0 (* x (* x (+ -0.5 (* (* x x) 0.20833333333333334)))))
   (/ 720.0 (* (* x x) (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0 + (x * (x * ((-0.5d0) + ((x * x) * 0.20833333333333334d0))))
    else
        tmp = 720.0d0 / ((x * x) * (x * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))))
	else:
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * 0.20833333333333334)))));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	else
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   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 64.5%

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

   5. Simplified 64.5%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. 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 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(720, \left({x}^{2} \cdot {x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(720, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 86.0%

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

   10. Simplified 86.0%

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

Alternative 11: 69.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.20833333333333334\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.42)
   (+ 1.0 (* x (* x (+ -0.5 (* (* x x) 0.20833333333333334)))))
   (/ 2.0 (* (* x x) (+ 1.0 (* (* x x) 0.08333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	} else {
		tmp = 2.0 / ((x * x) * (1.0 + ((x * x) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = 1.0d0 + (x * (x * ((-0.5d0) + ((x * x) * 0.20833333333333334d0))))
    else
        tmp = 2.0d0 / ((x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	} else {
		tmp = 2.0 / ((x * x) * (1.0 + ((x * x) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))))
	else:
		tmp = 2.0 / ((x * x) * (1.0 + ((x * x) * 0.08333333333333333)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * 0.20833333333333334)))));
	else
		tmp = Float64(2.0 / Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = 1.0 + (x * (x * (-0.5 + ((x * x) * 0.20833333333333334))));
	else
		tmp = 2.0 / ((x * x) * (1.0 + ((x * x) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.20833333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   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 64.5%

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

   5. Simplified 64.5%

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

   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

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
   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 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

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

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

      18. unpow2 N/A

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

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

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 75.3%

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

   8. Simplified 75.3%

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

Alternative 12: 69.4% 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 64.5%

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

   5. Simplified 64.5%

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

   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 75.3%

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

   5. Simplified 75.3%

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

   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 75.3%

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

   8. Simplified 75.3%

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

Alternative 13: 82.4% 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 82.3%

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

   5. Simplified 82.3%

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

   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 75.3%

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

   5. Simplified 75.3%

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

   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 75.3%

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

   8. Simplified 75.3%

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

4. Final simplification 80.6%

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

Alternative 14: 63.6% 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 64.7%

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

   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 61.2%

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

   5. Simplified 61.2%

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

   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 61.2%

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

   8. Simplified 61.2%

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

Alternative 15: 76.6% 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 77.3%

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

5. Simplified 77.3%

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

6. Final simplification 77.3%

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

Alternative 16: 51.2% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.0%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))