Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 15

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. cosh-def N/A

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ t_2 := x \cdot \left(x \cdot x + -2\right)\\ t_3 := x \cdot t\_2\\ \mathbf{if}\;x \leq 7 \cdot 10^{+30}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - t\_0 \cdot \left(t\_2 \cdot \left(t\_2 \cdot t\_2\right)\right)\right) \cdot \left(8 + \left(x \cdot x\right) \cdot t\_1\right)}{16 + t\_3 \cdot \left(t\_3 + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + x \cdot \left(x \cdot t\_1\right)\right) \cdot \left(4 - t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x t_0))
        (t_2 (* x (+ (* x x) -2.0)))
        (t_3 (* x t_2)))
   (if (<= x 7e+30)
     (/
      32.0
      (/
       (* (- 64.0 (* t_0 (* t_2 (* t_2 t_2)))) (+ 8.0 (* (* x x) t_1)))
       (+ 16.0 (* t_3 (+ t_3 4.0)))))
     (/ 32.0 (* (+ 8.0 (* x (* x t_1))) (- 4.0 t_3))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = x * ((x * x) + -2.0);
	double t_3 = x * t_2;
	double tmp;
	if (x <= 7e+30) {
		tmp = 32.0 / (((64.0 - (t_0 * (t_2 * (t_2 * t_2)))) * (8.0 + ((x * x) * t_1))) / (16.0 + (t_3 * (t_3 + 4.0))));
	} else {
		tmp = 32.0 / ((8.0 + (x * (x * t_1))) * (4.0 - t_3));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = x * t_0
    t_2 = x * ((x * x) + (-2.0d0))
    t_3 = x * t_2
    if (x <= 7d+30) then
        tmp = 32.0d0 / (((64.0d0 - (t_0 * (t_2 * (t_2 * t_2)))) * (8.0d0 + ((x * x) * t_1))) / (16.0d0 + (t_3 * (t_3 + 4.0d0))))
    else
        tmp = 32.0d0 / ((8.0d0 + (x * (x * t_1))) * (4.0d0 - t_3))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = x * ((x * x) + -2.0);
	double t_3 = x * t_2;
	double tmp;
	if (x <= 7e+30) {
		tmp = 32.0 / (((64.0 - (t_0 * (t_2 * (t_2 * t_2)))) * (8.0 + ((x * x) * t_1))) / (16.0 + (t_3 * (t_3 + 4.0))));
	} else {
		tmp = 32.0 / ((8.0 + (x * (x * t_1))) * (4.0 - t_3));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * t_0
	t_2 = x * ((x * x) + -2.0)
	t_3 = x * t_2
	tmp = 0
	if x <= 7e+30:
		tmp = 32.0 / (((64.0 - (t_0 * (t_2 * (t_2 * t_2)))) * (8.0 + ((x * x) * t_1))) / (16.0 + (t_3 * (t_3 + 4.0))))
	else:
		tmp = 32.0 / ((8.0 + (x * (x * t_1))) * (4.0 - t_3))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * Float64(Float64(x * x) + -2.0))
	t_3 = Float64(x * t_2)
	tmp = 0.0
	if (x <= 7e+30)
		tmp = Float64(32.0 / Float64(Float64(Float64(64.0 - Float64(t_0 * Float64(t_2 * Float64(t_2 * t_2)))) * Float64(8.0 + Float64(Float64(x * x) * t_1))) / Float64(16.0 + Float64(t_3 * Float64(t_3 + 4.0)))));
	else
		tmp = Float64(32.0 / Float64(Float64(8.0 + Float64(x * Float64(x * t_1))) * Float64(4.0 - t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = x * t_0;
	t_2 = x * ((x * x) + -2.0);
	t_3 = x * t_2;
	tmp = 0.0;
	if (x <= 7e+30)
		tmp = 32.0 / (((64.0 - (t_0 * (t_2 * (t_2 * t_2)))) * (8.0 + ((x * x) * t_1))) / (16.0 + (t_3 * (t_3 + 4.0))));
	else
		tmp = 32.0 / ((8.0 + (x * (x * t_1))) * (4.0 - t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$2), $MachinePrecision]}, If[LessEqual[x, 7e+30], N[(32.0 / N[(N[(N[(64.0 - N[(t$95$0 * N[(t$95$2 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 + N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(16.0 + N[(t$95$3 * N[(t$95$3 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(32.0 / N[(N[(8.0 + N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000042e30

   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 77.2%

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

   5. Simplified 77.2%

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

   7. Applied egg-rr 62.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 93.2%

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

       1. *-commutative N/A

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

       2. flip3-- N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 66.8%

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

   if 7.00000000000000042e30 < 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 52.0%

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

   5. Simplified 52.0%

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

   7. Applied egg-rr 2.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 100.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+30}:\\ \;\;\;\;\frac{32}{\frac{\left(64 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x + -2\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x + -2\right)\right) \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)\right)\right) \cdot \left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{16 + \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right) + 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{32}{\left(8 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x + -2\right)\\ t_1 := x \cdot t\_0\\ t_2 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{32}{\frac{\left(8 + \left(x \cdot x\right) \cdot t\_2\right) \cdot \left(16 - x \cdot \left(t\_0 \cdot t\_1\right)\right)}{t\_1 + 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ (* x x) -2.0))) (t_1 (* x t_0)) (t_2 (* x (* x (* x x)))))
   (if (<= x 1.15e+77)
     (/
      32.0
      (/ (* (+ 8.0 (* (* x x) t_2)) (- 16.0 (* x (* t_0 t_1)))) (+ t_1 4.0)))
     (/ 24.0 t_2))))

C

double code(double x) {
	double t_0 = x * ((x * x) + -2.0);
	double t_1 = x * t_0;
	double t_2 = x * (x * (x * x));
	double tmp;
	if (x <= 1.15e+77) {
		tmp = 32.0 / (((8.0 + ((x * x) * t_2)) * (16.0 - (x * (t_0 * t_1)))) / (t_1 + 4.0));
	} else {
		tmp = 24.0 / 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) + (-2.0d0))
    t_1 = x * t_0
    t_2 = x * (x * (x * x))
    if (x <= 1.15d+77) then
        tmp = 32.0d0 / (((8.0d0 + ((x * x) * t_2)) * (16.0d0 - (x * (t_0 * t_1)))) / (t_1 + 4.0d0))
    else
        tmp = 24.0d0 / t_2
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * ((x * x) + -2.0);
	double t_1 = x * t_0;
	double t_2 = x * (x * (x * x));
	double tmp;
	if (x <= 1.15e+77) {
		tmp = 32.0 / (((8.0 + ((x * x) * t_2)) * (16.0 - (x * (t_0 * t_1)))) / (t_1 + 4.0));
	} else {
		tmp = 24.0 / t_2;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * ((x * x) + -2.0)
	t_1 = x * t_0
	t_2 = x * (x * (x * x))
	tmp = 0
	if x <= 1.15e+77:
		tmp = 32.0 / (((8.0 + ((x * x) * t_2)) * (16.0 - (x * (t_0 * t_1)))) / (t_1 + 4.0))
	else:
		tmp = 24.0 / t_2
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(Float64(x * x) + -2.0))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 1.15e+77)
		tmp = Float64(32.0 / Float64(Float64(Float64(8.0 + Float64(Float64(x * x) * t_2)) * Float64(16.0 - Float64(x * Float64(t_0 * t_1)))) / Float64(t_1 + 4.0)));
	else
		tmp = Float64(24.0 / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * ((x * x) + -2.0);
	t_1 = x * t_0;
	t_2 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= 1.15e+77)
		tmp = 32.0 / (((8.0 + ((x * x) * t_2)) * (16.0 - (x * (t_0 * t_1)))) / (t_1 + 4.0));
	else
		tmp = 24.0 / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.15e+77], N[(32.0 / N[(N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(16.0 - N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(24.0 / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.14999999999999997e77

   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 74.2%

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

   5. Simplified 74.2%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 93.5%

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

       1. *-commutative N/A

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

       2. flip-- N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 71.9%

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

   if 1.14999999999999997e77 < 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

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{32}{\frac{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(16 - x \cdot \left(\left(x \cdot \left(x \cdot x + -2\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)\right)\right)}{x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right) + 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(32.0 / N[(N[(8.0 + N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(x * N[(x * N[(N[(x * x), $MachinePrecision] + -2.0), $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 72.4%

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

5. Simplified 72.4%

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

7. Applied egg-rr 51.0%

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

8. Taylor expanded in x around 0

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

10. Simplified 94.5%

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

Alternative 5: 91.9% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. cosh-def N/A

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. *-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. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

   14. *-lowering-*.f64 91.2%

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

7. Simplified 91.2%

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

Alternative 6: 89.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-288}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0)
   (/ 2.0 (+ 2.0 (* (* x x) (+ 1.0 (* (* x x) 0.08333333333333333)))))
   (/ (/ -288.0 (* x x)) (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = (-288.0 / (x * x)) / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0))))
    else
        tmp = ((-288.0d0) / (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) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = (-288.0 / (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) * (1.0 + ((x * x) * 0.08333333333333333))))
	else:
		tmp = (-288.0 / (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(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333)))));
	else
		tmp = Float64(Float64(-288.0 / 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) * (1.0 + ((x * x) * 0.08333333333333333))));
	else
		tmp = (-288.0 / (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[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-288.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $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 89.8%

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

   5. Simplified 89.8%

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

   if 700 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.8%

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24 - 288 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(24 - 288 \cdot \frac{1}{{x}^{2}}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(24 + \left(\mathsf{neg}\left(288 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(288 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(\frac{288 \cdot 1}{{x}^{2}}\right)\right)\right), \left({x}^{4}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(\frac{288}{{x}^{2}}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\frac{\mathsf{neg}\left(288\right)}{{x}^{2}}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(288\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{4}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \left({x}^{2}\right)\right)\right), \left({x}^{4}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \left(x \cdot x\right)\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{4}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 76.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   8. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{24 + \frac{-288}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-288}{{x}^{2}}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \left({x}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \left(x \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 85.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   11. Simplified 85.8%

       \[\leadsto \frac{\color{blue}{\frac{-288}{x \cdot x}}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.6% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. cosh-def N/A

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. *-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. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

   14. *-lowering-*.f64 91.2%

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

7. Simplified 91.2%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right)\right)\right) \]
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. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 91.2%

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

10. Simplified 91.2%

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

Alternative 8: 83.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{x \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-288}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0)
   (/ 2.0 (+ (* x x) 2.0))
   (/ (/ -288.0 (* x x)) (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / ((x * x) + 2.0);
	} else {
		tmp = (-288.0 / (x * x)) / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 2.0d0 / ((x * x) + 2.0d0)
    else
        tmp = ((-288.0d0) / (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 / ((x * x) + 2.0);
	} else {
		tmp = (-288.0 / (x * x)) / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 700.0:
		tmp = 2.0 / ((x * x) + 2.0)
	else:
		tmp = (-288.0 / (x * x)) / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 700.0)
		tmp = Float64(2.0 / Float64(Float64(x * x) + 2.0));
	else
		tmp = Float64(Float64(-288.0 / 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 / ((x * x) + 2.0);
	else
		tmp = (-288.0 / (x * x)) / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 700.0], N[(2.0 / N[(N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-288.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $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

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2}\right)}\right) \]
   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 78.7%

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

   5. Simplified 78.7%

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

   if 700 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.8%

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24 - 288 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(24 - 288 \cdot \frac{1}{{x}^{2}}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(24 + \left(\mathsf{neg}\left(288 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(288 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(\frac{288 \cdot 1}{{x}^{2}}\right)\right)\right), \left({x}^{4}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\mathsf{neg}\left(\frac{288}{{x}^{2}}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \left(\frac{\mathsf{neg}\left(288\right)}{{x}^{2}}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(288\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{4}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \left({x}^{2}\right)\right)\right), \left({x}^{4}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \left(x \cdot x\right)\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{4}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 76.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(24, \mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   8. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{24 + \frac{-288}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-288}{{x}^{2}}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \left({x}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \left(x \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 85.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-288, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   11. Simplified 85.8%

       \[\leadsto \frac{\color{blue}{\frac{-288}{x \cdot x}}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{x \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-288}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ 1.0 (* (* x x) -0.5))
   (/ 2.0 (* x (* x (+ 1.0 (* x (* x 0.08333333333333333))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 1.0 + ((x * x) * -0.5);
	} 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 <= 1.2d0) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    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 <= 1.2) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 2.0 / (x * (x * (1.0 + (x * (x * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(1.0 + N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
   4. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.6%

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

   5. Simplified 63.6%

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

   if 1.19999999999999996 < 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 75.8%

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

   5. Simplified 75.8%

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

   6. 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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 75.8%

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

   8. Simplified 75.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4) (+ 1.0 (* (* x x) -0.5)) (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0 + ((x * x) * -0.5);
	} 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 <= 1.4d0) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    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 <= 1.4) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 24.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	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 <= 1.4)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 24.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $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 < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
   4. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.6%

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

   5. Simplified 63.6%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)}\right) \]
   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 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.3% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 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 87.1%

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

5. Simplified 87.1%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 87.0%

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

8. Simplified 87.0%

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

Alternative 12: 63.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (+ 1.0 (* (* x x) -0.5)) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
   4. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.6%

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

   5. Simplified 63.6%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 47.7%

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

   5. Simplified 47.7%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 47.7%

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

   8. Simplified 47.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.4% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 63.9%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 47.7%

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

   5. Simplified 47.7%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 47.7%

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

   8. Simplified 47.7%

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

Alternative 14: 75.9% 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 72.4%

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

5. Simplified 72.4%

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

6. Final simplification 72.4%

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

Alternative 15: 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 51.1%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))