Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 12

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. rec-exp N/A

      \[\leadsto \frac{2}{e^{x} + \frac{1}{\color{blue}{e^{x}}}} \]

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + \frac{1}{e^{x}}}{2}}} \]

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

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

   4. rec-exp N/A

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

   5. cosh-def N/A

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

   6. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 74.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ t_2 := x \cdot \left(x \cdot t\_1\right)\\ \mathbf{if}\;x \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\frac{\frac{256 - t\_0 \cdot \left(t\_1 \cdot t\_1\right)}{16 + t\_2}}{\left(t\_0 + 4\right) \cdot \left(2 - x \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{16}{16 - t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))) (t_1 (* (* x x) t_0)) (t_2 (* x (* x t_1))))
   (if (<= x 3.4e+38)
     (/
      2.0
      (/
       (/ (- 256.0 (* t_0 (* t_1 t_1))) (+ 16.0 t_2))
       (* (+ t_0 4.0) (- 2.0 (* x x)))))
     (/ 16.0 (- 16.0 t_2)))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = x * (x * t_1);
	double tmp;
	if (x <= 3.4e+38) {
		tmp = 2.0 / (((256.0 - (t_0 * (t_1 * t_1))) / (16.0 + t_2)) / ((t_0 + 4.0) * (2.0 - (x * x))));
	} else {
		tmp = 16.0 / (16.0 - t_2);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = (x * x) * t_0
    t_2 = x * (x * t_1)
    if (x <= 3.4d+38) then
        tmp = 2.0d0 / (((256.0d0 - (t_0 * (t_1 * t_1))) / (16.0d0 + t_2)) / ((t_0 + 4.0d0) * (2.0d0 - (x * x))))
    else
        tmp = 16.0d0 / (16.0d0 - t_2)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = x * (x * t_1);
	double tmp;
	if (x <= 3.4e+38) {
		tmp = 2.0 / (((256.0 - (t_0 * (t_1 * t_1))) / (16.0 + t_2)) / ((t_0 + 4.0) * (2.0 - (x * x))));
	} else {
		tmp = 16.0 / (16.0 - t_2);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = (x * x) * t_0
	t_2 = x * (x * t_1)
	tmp = 0
	if x <= 3.4e+38:
		tmp = 2.0 / (((256.0 - (t_0 * (t_1 * t_1))) / (16.0 + t_2)) / ((t_0 + 4.0) * (2.0 - (x * x))))
	else:
		tmp = 16.0 / (16.0 - t_2)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(Float64(x * x) * t_0)
	t_2 = Float64(x * Float64(x * t_1))
	tmp = 0.0
	if (x <= 3.4e+38)
		tmp = Float64(2.0 / Float64(Float64(Float64(256.0 - Float64(t_0 * Float64(t_1 * t_1))) / Float64(16.0 + t_2)) / Float64(Float64(t_0 + 4.0) * Float64(2.0 - Float64(x * x)))));
	else
		tmp = Float64(16.0 / Float64(16.0 - t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	t_1 = (x * x) * t_0;
	t_2 = x * (x * t_1);
	tmp = 0.0;
	if (x <= 3.4e+38)
		tmp = 2.0 / (((256.0 - (t_0 * (t_1 * t_1))) / (16.0 + t_2)) / ((t_0 + 4.0) * (2.0 - (x * x))));
	else
		tmp = 16.0 / (16.0 - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.4e+38], N[(2.0 / N[(N[(N[(256.0 - N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(16.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + 4.0), $MachinePrecision] * N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(16.0 / N[(16.0 - t$95$2), $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 80.5%

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

   5. Simplified 80.5%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}} \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. flip-- N/A

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

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

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

   9. Applied egg-rr 65.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{256 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{16 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}}}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}} \]

   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 64.0%

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

   5. Simplified 64.0%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 7.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)\right)\right), \color{blue}{\left(\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1\right)}\right) \]

   9. Applied egg-rr 7.9%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 100.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\frac{\frac{256 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{16 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right) + 4\right) \cdot \left(2 - x \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{16}{16 - x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\frac{8 + \frac{t\_1 \cdot t\_1}{1728}}{4 + \frac{\frac{x \cdot x}{\frac{12}{x \cdot x}} - 2}{\frac{12}{t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{16}{16 - x \cdot \left(x \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))) (t_1 (* (* x x) t_0)))
   (if (<= x 5e+38)
     (/
      2.0
      (/
       (+ 8.0 (/ (* t_1 t_1) 1728.0))
       (+ 4.0 (/ (- (/ (* x x) (/ 12.0 (* x x))) 2.0) (/ 12.0 t_0)))))
     (/ 16.0 (- 16.0 (* x (* x t_1)))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 5e+38) {
		tmp = 2.0 / ((8.0 + ((t_1 * t_1) / 1728.0)) / (4.0 + ((((x * x) / (12.0 / (x * x))) - 2.0) / (12.0 / t_0))));
	} else {
		tmp = 16.0 / (16.0 - (x * (x * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = (x * x) * t_0
    if (x <= 5d+38) then
        tmp = 2.0d0 / ((8.0d0 + ((t_1 * t_1) / 1728.0d0)) / (4.0d0 + ((((x * x) / (12.0d0 / (x * x))) - 2.0d0) / (12.0d0 / t_0))))
    else
        tmp = 16.0d0 / (16.0d0 - (x * (x * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 5e+38) {
		tmp = 2.0 / ((8.0 + ((t_1 * t_1) / 1728.0)) / (4.0 + ((((x * x) / (12.0 / (x * x))) - 2.0) / (12.0 / t_0))));
	} else {
		tmp = 16.0 / (16.0 - (x * (x * t_1)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = (x * x) * t_0
	tmp = 0
	if x <= 5e+38:
		tmp = 2.0 / ((8.0 + ((t_1 * t_1) / 1728.0)) / (4.0 + ((((x * x) / (12.0 / (x * x))) - 2.0) / (12.0 / t_0))))
	else:
		tmp = 16.0 / (16.0 - (x * (x * t_1)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= 5e+38)
		tmp = Float64(2.0 / Float64(Float64(8.0 + Float64(Float64(t_1 * t_1) / 1728.0)) / Float64(4.0 + Float64(Float64(Float64(Float64(x * x) / Float64(12.0 / Float64(x * x))) - 2.0) / Float64(12.0 / t_0)))));
	else
		tmp = Float64(16.0 / Float64(16.0 - Float64(x * Float64(x * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	t_1 = (x * x) * t_0;
	tmp = 0.0;
	if (x <= 5e+38)
		tmp = 2.0 / ((8.0 + ((t_1 * t_1) / 1728.0)) / (4.0 + ((((x * x) / (12.0 / (x * x))) - 2.0) / (12.0 / t_0))));
	else
		tmp = 16.0 / (16.0 - (x * (x * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, 5e+38], N[(2.0 / N[(N[(8.0 + N[(N[(t$95$1 * t$95$1), $MachinePrecision] / 1728.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(N[(N[(N[(x * x), $MachinePrecision] / N[(12.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(12.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(16.0 / N[(16.0 - N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e38

   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 91.3%

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

   5. Simplified 91.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}}}\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\frac{x \cdot x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}\right)\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}}\right)\right)\right)\right) \]

      7. flip3-+ N/A

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

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

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

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

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

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

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

      11. *-lowering-*.f64 91.3%

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

   7. Applied egg-rr 91.3%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 91.0%

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

   10. Simplified 91.0%

       \[\leadsto \frac{2}{2 + \frac{x \cdot x}{\color{blue}{\frac{12}{x \cdot x}}}} \]
   11. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{2}^{3} + {\left(\frac{x \cdot x}{\frac{12}{x \cdot x}}\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\frac{x \cdot x}{\frac{12}{x \cdot x}} \cdot \frac{x \cdot x}{\frac{12}{x \cdot x}} - 2 \cdot \frac{x \cdot x}{\frac{12}{x \cdot x}}\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({2}^{3} + {\left(\frac{x \cdot x}{\frac{12}{x \cdot x}}\right)}^{3}\right), \color{blue}{\left(2 \cdot 2 + \left(\frac{x \cdot x}{\frac{12}{x \cdot x}} \cdot \frac{x \cdot x}{\frac{12}{x \cdot x}} - 2 \cdot \frac{x \cdot x}{\frac{12}{x \cdot x}}\right)\right)}\right)\right) \]

   12. Applied egg-rr 64.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{8 + \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1728}}{4 + \frac{\frac{x \cdot x}{\frac{12}{x \cdot x}} - 2}{\frac{12}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}}}} \]

   if 4.9999999999999997e38 < 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 64.0%

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

   5. Simplified 64.0%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

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

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

   7. Applied egg-rr 7.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)\right)\right), \color{blue}{\left(\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1\right)}\right) \]

   9. Applied egg-rr 7.9%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 100.0%

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

Alternative 4: 93.0% accurate, 10.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 16.0 (- 16.0 (* x (* x (* (* x x) (* x (* x (* x x)))))))))

C

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

Fortran

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

Java

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

Python

def code(x):
	return 16.0 / (16.0 - (x * (x * ((x * x) * (x * (x * (x * x)))))))

Julia

function code(x)
	return Float64(16.0 / Float64(16.0 - Float64(x * Float64(x * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
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 76.5%

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

5. Simplified 76.5%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

   3. flip-- N/A

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

   4. frac-times N/A

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

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

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

7. Applied egg-rr 49.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1}{\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)}}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(\left(4 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(2 - x \cdot x\right)\right)\right), \color{blue}{\left(\left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 1\right)}\right) \]

9. Applied egg-rr 49.5%

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

10. Taylor expanded in x around 0

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

12. Simplified 95.1%

    \[\leadsto \frac{\color{blue}{16}}{16 - x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \]
13. Add Preprocessing

Alternative 5: 91.3% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. rec-exp N/A

      \[\leadsto \frac{2}{e^{x} + \frac{1}{\color{blue}{e^{x}}}} \]

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x} + \frac{1}{e^{x}}}{2}}} \]

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

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

   4. rec-exp N/A

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

   5. cosh-def N/A

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

   6. 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 92.8%

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

7. Simplified 92.8%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 92.7%

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

10. Simplified 92.7%

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

Alternative 6: 68.9% 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 64.4%

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

   5. Simplified 64.4%

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

   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 79.5%

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

   5. Simplified 79.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      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 79.5%

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

   8. Simplified 79.5%

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

Alternative 7: 87.7% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 90.2%

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

5. Simplified 90.2%

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

Alternative 8: 81.9% accurate, 14.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 83.0%

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

   5. Simplified 83.0%

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

   if 3.7000000000000002 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 79.5%

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

   5. Simplified 79.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      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 79.5%

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

   8. Simplified 79.5%

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

Alternative 9: 87.3% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + \frac{x \cdot x}{\frac{12}{x \cdot x}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ 2.0 (/ (* x x) (/ 12.0 (* x x))))))

C

double code(double x) {
	return 2.0 / (2.0 + ((x * x) / (12.0 / (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + ((x * x) / (12.0d0 / (x * x))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + ((x * x) / (12.0 / (x * x))));
}

Python

def code(x):
	return 2.0 / (2.0 + ((x * x) / (12.0 / (x * x))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) / Float64(12.0 / Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + ((x * x) / (12.0 / (x * x))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] / N[(12.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 90.2%

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

5. Simplified 90.2%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}}}\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\frac{x \cdot x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}\right)\right)\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}}}\right)\right)\right)\right) \]

   7. flip3-+ N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f64 90.2%

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

7. Applied egg-rr 90.2%

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

8. Taylor expanded in x around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 90.0%

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

10. Simplified 90.0%

    \[\leadsto \frac{2}{2 + \frac{x \cdot x}{\color{blue}{\frac{12}{x \cdot x}}}} \]
11. Add Preprocessing

Alternative 10: 62.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.45) 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		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 <= 1.45)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.7%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
   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 58.8%

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

   5. Simplified 58.8%

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

   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 58.8%

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

   8. Simplified 58.8%

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

Alternative 11: 75.7% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 76.5%

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

5. Simplified 76.5%

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

Alternative 12: 49.9% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 48.1%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))