Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.9%

Time: 12.4s

Alternatives: 25

Speedup: 1.4×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ t_2 := J \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_1 \cdot t\_2 + U\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\ell, t\_2 \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(t\_0 - e^{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)) (t_2 (* J (cos (* 0.5 K)))))
   (if (<= t_1 -0.5)
     (+ (* t_1 t_2) U)
     (if (<= t_1 1e-9)
       (fma l (* t_2 (fma 0.3333333333333333 (pow l 2.0) 2.0)) U)
       (- U (* (cos (/ K 2.0)) (* J (- t_0 (exp l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double t_2 = J * cos((0.5 * K));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (t_1 * t_2) + U;
	} else if (t_1 <= 1e-9) {
		tmp = fma(l, (t_2 * fma(0.3333333333333333, pow(l, 2.0), 2.0)), U);
	} else {
		tmp = U - (cos((K / 2.0)) * (J * (t_0 - exp(l))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	t_2 = Float64(J * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(t_1 * t_2) + U);
	elseif (t_1 <= 1e-9)
		tmp = fma(l, Float64(t_2 * fma(0.3333333333333333, (l ^ 2.0), 2.0)), U);
	else
		tmp = Float64(U - Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(t_0 - exp(l)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(t$95$1 * t$95$2), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(l * N[(t$95$2 * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U - N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(t$95$0 - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.5

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -0.5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000006e-9

   1. Initial program 66.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{U + \ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U} \]

      2. fma-define 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right)} \]

      3. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot 0.3333333333333333} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right) \]

      4. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{2}\right)}\right) \cdot 0.3333333333333333 + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot {\ell}^{2}\right)} \cdot 0.3333333333333333 + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right) \]

      6. associate-*r* 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{2} \cdot 0.3333333333333333\right)} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right) \]

      7. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right), U\right) \]

      8. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{2}\right) + \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 2}, U\right) \]

      9. distribute-lft-out 99.9%

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{2} + 2\right)}, U\right) \]

      10. fma-define 99.9%

          \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)}, U\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right), U\right)} \]

   if 1.00000000000000006e-9 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.5:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{-\ell} - e^{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.5 \lor \neg \left(t\_1 \leq 10^{-9}\right):\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(t\_0 - e^{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)))
   (if (or (<= t_1 -0.5) (not (<= t_1 1e-9)))
     (- U (* (cos (/ K 2.0)) (* J (- t_0 (exp l)))))
     (+ U (* (* J (cos (* 0.5 K))) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double tmp;
	if ((t_1 <= -0.5) || !(t_1 <= 1e-9)) {
		tmp = U - (cos((K / 2.0)) * (J * (t_0 - exp(l))));
	} else {
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = exp(l) - t_0
    if ((t_1 <= (-0.5d0)) .or. (.not. (t_1 <= 1d-9))) then
        tmp = u - (cos((k / 2.0d0)) * (j * (t_0 - exp(l))))
    else
        tmp = u + ((j * cos((0.5d0 * k))) * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.exp(l) - t_0;
	double tmp;
	if ((t_1 <= -0.5) || !(t_1 <= 1e-9)) {
		tmp = U - (Math.cos((K / 2.0)) * (J * (t_0 - Math.exp(l))));
	} else {
		tmp = U + ((J * Math.cos((0.5 * K))) * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.exp(l) - t_0
	tmp = 0
	if (t_1 <= -0.5) or not (t_1 <= 1e-9):
		tmp = U - (math.cos((K / 2.0)) * (J * (t_0 - math.exp(l))))
	else:
		tmp = U + ((J * math.cos((0.5 * K))) * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	tmp = 0.0
	if ((t_1 <= -0.5) || !(t_1 <= 1e-9))
		tmp = Float64(U - Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(t_0 - exp(l)))));
	else
		tmp = Float64(U + Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = exp(l) - t_0;
	tmp = 0.0;
	if ((t_1 <= -0.5) || ~((t_1 <= 1e-9)))
		tmp = U - (cos((K / 2.0)) * (J * (t_0 - exp(l))));
	else
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.5], N[Not[LessEqual[t$95$1, 1e-9]], $MachinePrecision]], N[(U - N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(t$95$0 - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.5 or 1.00000000000000006e-9 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   if -0.5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000006e-9

   1. Initial program 66.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 66.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 66.9%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 66.9%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.5 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 10^{-9}\right):\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{-\ell} - e^{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ t_2 := J \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_1 \cdot t\_2 + U\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;U + t\_2 \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(t\_0 - e^{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)) (t_2 (* J (cos (* 0.5 K)))))
   (if (<= t_1 -0.5)
     (+ (* t_1 t_2) U)
     (if (<= t_1 1e-9)
       (+ U (* t_2 (* l 2.0)))
       (- U (* (cos (/ K 2.0)) (* J (- t_0 (exp l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double t_2 = J * cos((0.5 * K));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (t_1 * t_2) + U;
	} else if (t_1 <= 1e-9) {
		tmp = U + (t_2 * (l * 2.0));
	} else {
		tmp = U - (cos((K / 2.0)) * (J * (t_0 - exp(l))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = exp(l) - t_0
    t_2 = j * cos((0.5d0 * k))
    if (t_1 <= (-0.5d0)) then
        tmp = (t_1 * t_2) + u
    else if (t_1 <= 1d-9) then
        tmp = u + (t_2 * (l * 2.0d0))
    else
        tmp = u - (cos((k / 2.0d0)) * (j * (t_0 - exp(l))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.exp(l) - t_0;
	double t_2 = J * Math.cos((0.5 * K));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (t_1 * t_2) + U;
	} else if (t_1 <= 1e-9) {
		tmp = U + (t_2 * (l * 2.0));
	} else {
		tmp = U - (Math.cos((K / 2.0)) * (J * (t_0 - Math.exp(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.exp(l) - t_0
	t_2 = J * math.cos((0.5 * K))
	tmp = 0
	if t_1 <= -0.5:
		tmp = (t_1 * t_2) + U
	elif t_1 <= 1e-9:
		tmp = U + (t_2 * (l * 2.0))
	else:
		tmp = U - (math.cos((K / 2.0)) * (J * (t_0 - math.exp(l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	t_2 = Float64(J * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(t_1 * t_2) + U);
	elseif (t_1 <= 1e-9)
		tmp = Float64(U + Float64(t_2 * Float64(l * 2.0)));
	else
		tmp = Float64(U - Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(t_0 - exp(l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = exp(l) - t_0;
	t_2 = J * cos((0.5 * K));
	tmp = 0.0;
	if (t_1 <= -0.5)
		tmp = (t_1 * t_2) + U;
	elseif (t_1 <= 1e-9)
		tmp = U + (t_2 * (l * 2.0));
	else
		tmp = U - (cos((K / 2.0)) * (J * (t_0 - exp(l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(t$95$1 * t$95$2), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(U + N[(t$95$2 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U - N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(t$95$0 - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.5

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -0.5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000006e-9

   1. Initial program 66.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 66.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 66.9%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 66.9%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   if 1.00000000000000006e-9 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.5:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 10^{-9}:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U - \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{-\ell} - e^{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.25:\\ \;\;\;\;U + \left(J \cdot t\_0\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq -0.004:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + -0.25 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.001:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.25)
     (+ U (* (* J t_0) (* l 2.0)))
     (if (<= t_1 -0.004)
       (+ U (* J (* l (+ 2.0 (* -0.25 (pow K 2.0))))))
       (if (<= t_1 -0.001)
         (* J (* 2.0 (* l t_0)))
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.25) {
		tmp = U + ((J * t_0) * (l * 2.0));
	} else if (t_1 <= -0.004) {
		tmp = U + (J * (l * (2.0 + (-0.25 * pow(K, 2.0)))));
	} else if (t_1 <= -0.001) {
		tmp = J * (2.0 * (l * t_0));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((0.5d0 * k))
    t_1 = cos((k / 2.0d0))
    if (t_1 <= (-0.25d0)) then
        tmp = u + ((j * t_0) * (l * 2.0d0))
    else if (t_1 <= (-0.004d0)) then
        tmp = u + (j * (l * (2.0d0 + ((-0.25d0) * (k ** 2.0d0)))))
    else if (t_1 <= (-0.001d0)) then
        tmp = j * (2.0d0 * (l * t_0))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.25) {
		tmp = U + ((J * t_0) * (l * 2.0));
	} else if (t_1 <= -0.004) {
		tmp = U + (J * (l * (2.0 + (-0.25 * Math.pow(K, 2.0)))));
	} else if (t_1 <= -0.001) {
		tmp = J * (2.0 * (l * t_0));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -0.25:
		tmp = U + ((J * t_0) * (l * 2.0))
	elif t_1 <= -0.004:
		tmp = U + (J * (l * (2.0 + (-0.25 * math.pow(K, 2.0)))))
	elif t_1 <= -0.001:
		tmp = J * (2.0 * (l * t_0))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.25)
		tmp = Float64(U + Float64(Float64(J * t_0) * Float64(l * 2.0)));
	elseif (t_1 <= -0.004)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(-0.25 * (K ^ 2.0))))));
	elseif (t_1 <= -0.001)
		tmp = Float64(J * Float64(2.0 * Float64(l * t_0)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((0.5 * K));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -0.25)
		tmp = U + ((J * t_0) * (l * 2.0));
	elseif (t_1 <= -0.004)
		tmp = U + (J * (l * (2.0 + (-0.25 * (K ^ 2.0)))));
	elseif (t_1 <= -0.001)
		tmp = J * (2.0 * (l * t_0));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.25], N[(U + N[(N[(J * t$95$0), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.004], N[(U + N[(J * N[(l * N[(2.0 + N[(-0.25 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.001], N[(J * N[(2.0 * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.25

   1. Initial program 77.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 77.4%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 77.4%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 77.4%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 77.3%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   if -0.25 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 35.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 35.9%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 35.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 35.9%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 35.9%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 35.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 35.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 45.9%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in K around 0 66.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in 66.9%

         \[\leadsto -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \color{blue}{\left(J \cdot \left(2 \cdot \ell\right) + J \cdot \frac{U}{J}\right)} \]

      2. associate-+r+ 66.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + J \cdot \left(2 \cdot \ell\right)\right) + J \cdot \frac{U}{J}} \]

      3. *-commutative 66.9%

         \[\leadsto \left(\color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25} + J \cdot \left(2 \cdot \ell\right)\right) + J \cdot \frac{U}{J} \]

      4. associate-*l* 66.9%

         \[\leadsto \left(\color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot -0.25\right)} + J \cdot \left(2 \cdot \ell\right)\right) + J \cdot \frac{U}{J} \]

      5. associate-*r* 66.9%

         \[\leadsto \left(J \cdot \color{blue}{\left({K}^{2} \cdot \left(\ell \cdot -0.25\right)\right)} + J \cdot \left(2 \cdot \ell\right)\right) + J \cdot \frac{U}{J} \]

      6. *-commutative 66.9%

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot \color{blue}{\left(-0.25 \cdot \ell\right)}\right) + J \cdot \left(2 \cdot \ell\right)\right) + J \cdot \frac{U}{J} \]

      7. distribute-lft-out 78.0%

         \[\leadsto \color{blue}{J \cdot \left({K}^{2} \cdot \left(-0.25 \cdot \ell\right) + 2 \cdot \ell\right)} + J \cdot \frac{U}{J} \]

      8. +-commutative 78.0%

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell + {K}^{2} \cdot \left(-0.25 \cdot \ell\right)\right)} + J \cdot \frac{U}{J} \]

      9. associate-*r* 78.0%

         \[\leadsto J \cdot \left(2 \cdot \ell + \color{blue}{\left({K}^{2} \cdot -0.25\right) \cdot \ell}\right) + J \cdot \frac{U}{J} \]

      10. *-commutative 78.0%

          \[\leadsto J \cdot \left(2 \cdot \ell + \color{blue}{\left(-0.25 \cdot {K}^{2}\right)} \cdot \ell\right) + J \cdot \frac{U}{J} \]

      11. distribute-rgt-out 78.0%

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + -0.25 \cdot {K}^{2}\right)\right)} + J \cdot \frac{U}{J} \]

      12. *-commutative 78.0%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{K}^{2} \cdot -0.25}\right)\right) + J \cdot \frac{U}{J} \]

      13. associate-/l* 78.8%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + \color{blue}{\frac{J \cdot U}{J}} \]

      14. *-commutative 78.8%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + \frac{\color{blue}{U \cdot J}}{J} \]

      15. associate-/l* 89.2%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + \color{blue}{U \cdot \frac{J}{J}} \]

      16. *-inverses 89.2%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + U \cdot \color{blue}{1} \]

      17. *-rgt-identity 89.2%

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + \color{blue}{U} \]

   9. Simplified 89.2%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right) + U} \]

   if -0.0040000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1e-3

   1. Initial program 1.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 98.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 98.4%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 100.0%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in l around inf 100.0%

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

   if -1e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 88.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 84.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.25:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.004:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + -0.25 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -40:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.00042:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_1 (- U (* J (- (exp (- l)) (exp l))))))
   (if (<= l -1.02e+120)
     t_0
     (if (<= l -40.0)
       t_1
       (if (<= l 0.00042)
         (+ U (* (* J (cos (* 0.5 K))) (* l 2.0)))
         (if (<= l 4e+91) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U - (J * (exp(-l) - exp(l)));
	double tmp;
	if (l <= -1.02e+120) {
		tmp = t_0;
	} else if (l <= -40.0) {
		tmp = t_1;
	} else if (l <= 0.00042) {
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	} else if (l <= 4e+91) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_1 = u - (j * (exp(-l) - exp(l)))
    if (l <= (-1.02d+120)) then
        tmp = t_0
    else if (l <= (-40.0d0)) then
        tmp = t_1
    else if (l <= 0.00042d0) then
        tmp = u + ((j * cos((0.5d0 * k))) * (l * 2.0d0))
    else if (l <= 4d+91) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U - (J * (Math.exp(-l) - Math.exp(l)));
	double tmp;
	if (l <= -1.02e+120) {
		tmp = t_0;
	} else if (l <= -40.0) {
		tmp = t_1;
	} else if (l <= 0.00042) {
		tmp = U + ((J * Math.cos((0.5 * K))) * (l * 2.0));
	} else if (l <= 4e+91) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_1 = U - (J * (math.exp(-l) - math.exp(l)))
	tmp = 0
	if l <= -1.02e+120:
		tmp = t_0
	elif l <= -40.0:
		tmp = t_1
	elif l <= 0.00042:
		tmp = U + ((J * math.cos((0.5 * K))) * (l * 2.0))
	elif l <= 4e+91:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_1 = Float64(U - Float64(J * Float64(exp(Float64(-l)) - exp(l))))
	tmp = 0.0
	if (l <= -1.02e+120)
		tmp = t_0;
	elseif (l <= -40.0)
		tmp = t_1;
	elseif (l <= 0.00042)
		tmp = Float64(U + Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(l * 2.0)));
	elseif (l <= 4e+91)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_1 = U - (J * (exp(-l) - exp(l)));
	tmp = 0.0;
	if (l <= -1.02e+120)
		tmp = t_0;
	elseif (l <= -40.0)
		tmp = t_1;
	elseif (l <= 0.00042)
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	elseif (l <= 4e+91)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U - N[(J * N[(N[Exp[(-l)], $MachinePrecision] - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.02e+120], t$95$0, If[LessEqual[l, -40.0], t$95$1, If[LessEqual[l, 0.00042], N[(U + N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+91], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.01999999999999997e120 or 4.00000000000000032e91 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\left({\ell}^{3} \cdot J\right)} \cdot 0.3333333333333333\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.01999999999999997e120 < l < -40 or 4.2000000000000002e-4 < l < 4.00000000000000032e91

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 80.0%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -40 < l < 4.2000000000000002e-4

   1. Initial program 67.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 67.7%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 67.7%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 67.7%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 98.6%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+120}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -40:\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.00042:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+91}:\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t\_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -40:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.00046:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_2 (- U (* J (- (exp (- l)) (exp l))))))
   (if (<= l -1.02e+120)
     t_1
     (if (<= l -40.0)
       t_2
       (if (<= l 0.00046)
         (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 4e+91) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U - (J * (exp(-l) - exp(l)));
	double tmp;
	if (l <= -1.02e+120) {
		tmp = t_1;
	} else if (l <= -40.0) {
		tmp = t_2;
	} else if (l <= 0.00046) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 4e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_2 = u - (j * (exp(-l) - exp(l)))
    if (l <= (-1.02d+120)) then
        tmp = t_1
    else if (l <= (-40.0d0)) then
        tmp = t_2
    else if (l <= 0.00046d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 4d+91) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U - (J * (Math.exp(-l) - Math.exp(l)));
	double tmp;
	if (l <= -1.02e+120) {
		tmp = t_1;
	} else if (l <= -40.0) {
		tmp = t_2;
	} else if (l <= 0.00046) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 4e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_2 = U - (J * (math.exp(-l) - math.exp(l)))
	tmp = 0
	if l <= -1.02e+120:
		tmp = t_1
	elif l <= -40.0:
		tmp = t_2
	elif l <= 0.00046:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 4e+91:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_2 = Float64(U - Float64(J * Float64(exp(Float64(-l)) - exp(l))))
	tmp = 0.0
	if (l <= -1.02e+120)
		tmp = t_1;
	elseif (l <= -40.0)
		tmp = t_2;
	elseif (l <= 0.00046)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 4e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_2 = U - (J * (exp(-l) - exp(l)));
	tmp = 0.0;
	if (l <= -1.02e+120)
		tmp = t_1;
	elseif (l <= -40.0)
		tmp = t_2;
	elseif (l <= 0.00046)
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 4e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U - N[(J * N[(N[Exp[(-l)], $MachinePrecision] - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.02e+120], t$95$1, If[LessEqual[l, -40.0], t$95$2, If[LessEqual[l, 0.00046], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+91], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.01999999999999997e120 or 4.00000000000000032e91 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\left({\ell}^{3} \cdot J\right)} \cdot 0.3333333333333333\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.01999999999999997e120 < l < -40 or 4.6000000000000001e-4 < l < 4.00000000000000032e91

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 80.0%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -40 < l < 4.6000000000000001e-4

   1. Initial program 67.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 98.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+120}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -40:\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.00046:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+91}:\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;J \cdot \left(U \cdot \left(\frac{1}{J} - -2 \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.001)
   (* J (* U (- (/ 1.0 J) (* -2.0 (* l (/ (cos (* 0.5 K)) U))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.001) {
		tmp = J * (U * ((1.0 / J) - (-2.0 * (l * (cos((0.5 * K)) / U)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.001d0)) then
        tmp = j * (u * ((1.0d0 / j) - ((-2.0d0) * (l * (cos((0.5d0 * k)) / u)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.001) {
		tmp = J * (U * ((1.0 / J) - (-2.0 * (l * (Math.cos((0.5 * K)) / U)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.001:
		tmp = J * (U * ((1.0 / J) - (-2.0 * (l * (math.cos((0.5 * K)) / U)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.001)
		tmp = Float64(J * Float64(U * Float64(Float64(1.0 / J) - Float64(-2.0 * Float64(l * Float64(cos(Float64(0.5 * K)) / U))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.001)
		tmp = J * (U * ((1.0 / J) - (-2.0 * (l * (cos((0.5 * K)) / U)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.001], N[(J * N[(U * N[(N[(1.0 / J), $MachinePrecision] - N[(-2.0 * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1e-3

   1. Initial program 79.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 71.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 71.5%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 71.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 71.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 71.6%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 71.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 73.0%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in U around -inf 76.1%

      \[\leadsto J \cdot \color{blue}{\left(-1 \cdot \left(U \cdot \left(-2 \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U} - \frac{1}{J}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 76.1%

         \[\leadsto J \cdot \color{blue}{\left(\left(-1 \cdot U\right) \cdot \left(-2 \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U} - \frac{1}{J}\right)\right)} \]

      2. neg-mul-1 76.1%

         \[\leadsto J \cdot \left(\color{blue}{\left(-U\right)} \cdot \left(-2 \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U} - \frac{1}{J}\right)\right) \]

      3. associate-/l* 76.1%

         \[\leadsto J \cdot \left(\left(-U\right) \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)} - \frac{1}{J}\right)\right) \]

   9. Simplified 76.1%

      \[\leadsto J \cdot \color{blue}{\left(\left(-U\right) \cdot \left(-2 \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right) - \frac{1}{J}\right)\right)} \]

   if -1e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 88.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 84.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;J \cdot \left(U \cdot \left(\frac{1}{J} - -2 \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.33:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.33)
   (* U (+ 1.0 (* 2.0 (* J (* l (/ (cos (* 0.5 K)) U))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.33) {
		tmp = U * (1.0 + (2.0 * (J * (l * (cos((0.5 * K)) / U)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.33d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l * (cos((0.5d0 * k)) / u)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.33) {
		tmp = U * (1.0 + (2.0 * (J * (l * (Math.cos((0.5 * K)) / U)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.33:
		tmp = U * (1.0 + (2.0 * (J * (l * (math.cos((0.5 * K)) / U)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.33)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l * Float64(cos(Float64(0.5 * K)) / U))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.33)
		tmp = U * (1.0 + (2.0 * (J * (l * (cos((0.5 * K)) / U)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.33], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.330000000000000016

   1. Initial program 78.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 70.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 70.9%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 70.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 70.9%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 70.9%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 69.5%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in U around inf 70.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   8. Step-by-step derivation

      1. associate-/l* 73.6%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

      2. associate-/l* 73.6%

         \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)}\right)\right) \]

   9. Simplified 73.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right)\right)} \]

   if 0.330000000000000016 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 88.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 85.3%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.33:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.001)
   (* J (+ (* 2.0 (* l (cos (* 0.5 K)))) (/ U J)))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.001) {
		tmp = J * ((2.0 * (l * cos((0.5 * K)))) + (U / J));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.001d0)) then
        tmp = j * ((2.0d0 * (l * cos((0.5d0 * k)))) + (u / j))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.001) {
		tmp = J * ((2.0 * (l * Math.cos((0.5 * K)))) + (U / J));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.001:
		tmp = J * ((2.0 * (l * math.cos((0.5 * K)))) + (U / J))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.001)
		tmp = Float64(J * Float64(Float64(2.0 * Float64(l * cos(Float64(0.5 * K)))) + Float64(U / J)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.001)
		tmp = J * ((2.0 * (l * cos((0.5 * K)))) + (U / J));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.001], N[(J * N[(N[(2.0 * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1e-3

   1. Initial program 79.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 71.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 71.5%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 71.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 71.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 71.6%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 71.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 73.0%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   if -1e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 88.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 84.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;U + \cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.001)
   (+ U (* (cos (* 0.5 K)) (* l (* J 2.0))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.001) {
		tmp = U + (cos((0.5 * K)) * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.001d0)) then
        tmp = u + (cos((0.5d0 * k)) * (l * (j * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.001) {
		tmp = U + (Math.cos((0.5 * K)) * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.001:
		tmp = U + (math.cos((0.5 * K)) * (l * (J * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.001)
		tmp = Float64(U + Float64(cos(Float64(0.5 * K)) * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.001)
		tmp = U + (cos((0.5 * K)) * (l * (J * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.001], N[(U + N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1e-3

   1. Initial program 79.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 71.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 71.5%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 71.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 71.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 71.6%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 71.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   if -1e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 88.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 84.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.001:\\ \;\;\;\;U + \cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -40 \lor \neg \left(\ell \leq 0.00031\right):\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -40.0) (not (<= l 0.00031)))
   (- U (* J (- (exp (- l)) (exp l))))
   (+ U (* (* J (cos (* 0.5 K))) (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -40.0) || !(l <= 0.00031)) {
		tmp = U - (J * (exp(-l) - exp(l)));
	} else {
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-40.0d0)) .or. (.not. (l <= 0.00031d0))) then
        tmp = u - (j * (exp(-l) - exp(l)))
    else
        tmp = u + ((j * cos((0.5d0 * k))) * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -40.0) || !(l <= 0.00031)) {
		tmp = U - (J * (Math.exp(-l) - Math.exp(l)));
	} else {
		tmp = U + ((J * Math.cos((0.5 * K))) * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -40.0) or not (l <= 0.00031):
		tmp = U - (J * (math.exp(-l) - math.exp(l)))
	else:
		tmp = U + ((J * math.cos((0.5 * K))) * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -40.0) || !(l <= 0.00031))
		tmp = Float64(U - Float64(J * Float64(exp(Float64(-l)) - exp(l))));
	else
		tmp = Float64(U + Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -40.0) || ~((l <= 0.00031)))
		tmp = U - (J * (exp(-l) - exp(l)));
	else
		tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -40.0], N[Not[LessEqual[l, 0.00031]], $MachinePrecision]], N[(U - N[(J * N[(N[Exp[(-l)], $MachinePrecision] - N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -40 or 3.1e-4 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 81.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -40 < l < 3.1e-4

   1. Initial program 67.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 67.7%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 67.7%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

      2. *-commutative 67.7%

         \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 98.6%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -40 \lor \neg \left(\ell \leq 0.00031\right):\\ \;\;\;\;U - J \cdot \left(e^{-\ell} - e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -4.4 \cdot 10^{+205} \lor \neg \left(J \leq 4.7 \cdot 10^{+82}\right):\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -4.4e+205) (not (<= J 4.7e+82)))
   (* J (* 2.0 (* l (cos (* 0.5 K)))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -4.4e+205) || !(J <= 4.7e+82)) {
		tmp = J * (2.0 * (l * cos((0.5 * K))));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j <= (-4.4d+205)) .or. (.not. (j <= 4.7d+82))) then
        tmp = j * (2.0d0 * (l * cos((0.5d0 * k))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -4.4e+205) || !(J <= 4.7e+82)) {
		tmp = J * (2.0 * (l * Math.cos((0.5 * K))));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -4.4e+205) or not (J <= 4.7e+82):
		tmp = J * (2.0 * (l * math.cos((0.5 * K))))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -4.4e+205) || !(J <= 4.7e+82))
		tmp = Float64(J * Float64(2.0 * Float64(l * cos(Float64(0.5 * K)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -4.4e+205) || ~((J <= 4.7e+82)))
		tmp = J * (2.0 * (l * cos((0.5 * K))));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -4.4e+205], N[Not[LessEqual[J, 4.7e+82]], $MachinePrecision]], N[(J * N[(2.0 * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -4.3999999999999997e205 or 4.7e82 < J

   1. Initial program 70.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 84.0%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 84.0%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 84.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 84.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 84.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 84.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 84.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 84.0%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in l around inf 73.9%

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

   if -4.3999999999999997e205 < J < 4.7e82

   1. Initial program 90.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 56.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 56.3%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 56.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 56.3%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 56.3%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 56.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in K around 0 51.7%

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.4 \cdot 10^{+205} \lor \neg \left(J \leq 4.7 \cdot 10^{+82}\right):\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;J \leq -9.2 \cdot 10^{+208}:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;J \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t\_0 \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))))
   (if (<= J -9.2e+208)
     (* J (* 2.0 (* l t_0)))
     (if (<= J 4.7e+82) (+ U (* 2.0 (* l J))) (* l (* t_0 (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double tmp;
	if (J <= -9.2e+208) {
		tmp = J * (2.0 * (l * t_0));
	} else if (J <= 4.7e+82) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = l * (t_0 * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((0.5d0 * k))
    if (j <= (-9.2d+208)) then
        tmp = j * (2.0d0 * (l * t_0))
    else if (j <= 4.7d+82) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = l * (t_0 * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double tmp;
	if (J <= -9.2e+208) {
		tmp = J * (2.0 * (l * t_0));
	} else if (J <= 4.7e+82) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = l * (t_0 * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	tmp = 0
	if J <= -9.2e+208:
		tmp = J * (2.0 * (l * t_0))
	elif J <= 4.7e+82:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = l * (t_0 * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (J <= -9.2e+208)
		tmp = Float64(J * Float64(2.0 * Float64(l * t_0)));
	elseif (J <= 4.7e+82)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = Float64(l * Float64(t_0 * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((0.5 * K));
	tmp = 0.0;
	if (J <= -9.2e+208)
		tmp = J * (2.0 * (l * t_0));
	elseif (J <= 4.7e+82)
		tmp = U + (2.0 * (l * J));
	else
		tmp = l * (t_0 * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[J, -9.2e+208], N[(J * N[(2.0 * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.7e+82], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -9.20000000000000002e208

   1. Initial program 66.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 80.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 80.5%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 80.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 80.5%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 80.5%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 80.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 80.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 80.5%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   7. Taylor expanded in l around inf 80.5%

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

   if -9.20000000000000002e208 < J < 4.7e82

   1. Initial program 90.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 56.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 56.3%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 56.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 56.3%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 56.3%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 56.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in K around 0 51.7%

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]

   if 4.7e82 < J

   1. Initial program 72.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 85.6%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 85.6%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 85.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 85.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 85.6%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 85.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in l around inf 76.6%

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{\ell}\right)} \]

   7. Taylor expanded in J around inf 71.1%

      \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 71.1%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]

      2. *-commutative 71.1%

         \[\leadsto \ell \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) \]

   9. Simplified 71.1%

      \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.2 \cdot 10^{+208}:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;J \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))

C

double code(double J, double l, double K, double U) {
	return U + (2.0 * (J * (l * cos((0.5 * K)))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
}

Python

def code(J, l, K, U):
	return U + (2.0 * (J * (l * math.cos((0.5 * K)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

4. Final simplification 64.5%

   \[\leadsto U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) \]
5. Add Preprocessing

Alternative 15: 64.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* (* J (cos (* 0.5 K))) (* l 2.0))))

C

double code(double J, double l, double K, double U) {
	return U + ((J * cos((0.5 * K))) * (l * 2.0));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + ((j * cos((0.5d0 * k))) * (l * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + ((J * Math.cos((0.5 * K))) * (l * 2.0));
}

Python

def code(J, l, K, U):
	return U + ((J * math.cos((0.5 * K))) * (l * 2.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(l * 2.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + ((J * cos((0.5 * K))) * (l * 2.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0 84.5%

   \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 84.5%

      \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   2. *-commutative 84.5%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

5. Simplified 84.5%

   \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

6. Taylor expanded in l around 0 64.5%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

7. Final simplification 64.5%

   \[\leadsto U + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right) \]
8. Add Preprocessing

Alternative 16: 55.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \ell \cdot 2, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma J (* l 2.0) U))

C

double code(double J, double l, double K, double U) {
	return fma(J, (l * 2.0), U);
}

Julia

function code(J, l, K, U)
	return fma(J, Float64(l * 2.0), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   2. *-commutative 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

   3. associate-*r* 64.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

   4. *-commutative 64.5%

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

5. Simplified 64.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

6. Taylor expanded in K around 0 53.3%

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

   1. +-commutative 53.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

   2. *-commutative 53.3%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

   3. associate-*r* 53.3%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   4. fma-define 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

   5. *-commutative 53.3%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

8. Simplified 53.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

9. Final simplification 53.3%

   \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U\right) \]
10. Add Preprocessing

Alternative 17: 46.1% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -23500 \lor \neg \left(\ell \leq 680000000000\right):\\ \;\;\;\;J \cdot \left(J \cdot \left(U \cdot U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -23500.0) (not (<= l 680000000000.0))) (* J (* J (* U U))) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23500.0) || !(l <= 680000000000.0)) {
		tmp = J * (J * (U * U));
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-23500.0d0)) .or. (.not. (l <= 680000000000.0d0))) then
        tmp = j * (j * (u * u))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23500.0) || !(l <= 680000000000.0)) {
		tmp = J * (J * (U * U));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -23500.0) or not (l <= 680000000000.0):
		tmp = J * (J * (U * U))
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -23500.0) || !(l <= 680000000000.0))
		tmp = Float64(J * Float64(J * Float64(U * U)));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -23500.0) || ~((l <= 680000000000.0)))
		tmp = J * (J * (U * U));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -23500.0], N[Not[LessEqual[l, 680000000000.0]], $MachinePrecision]], N[(J * N[(J * N[(U * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -23500 or 6.8e11 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 33.0%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 33.0%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 33.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 33.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 33.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 33.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 33.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 39.5%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   8. Applied egg-rr 23.4%

      \[\leadsto J \cdot \color{blue}{\left(U \cdot \left(J \cdot U\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 23.4%

         \[\leadsto J \cdot \color{blue}{\left(\left(J \cdot U\right) \cdot U\right)} \]

      2. associate-*l* 25.6%

         \[\leadsto J \cdot \color{blue}{\left(J \cdot \left(U \cdot U\right)\right)} \]

   10. Simplified 25.6%

       \[\leadsto J \cdot \color{blue}{\left(J \cdot \left(U \cdot U\right)\right)} \]

   if -23500 < l < 6.8e11

   1. Initial program 68.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 64.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -23500 \lor \neg \left(\ell \leq 680000000000\right):\\ \;\;\;\;J \cdot \left(J \cdot \left(U \cdot U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.4% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -23500:\\ \;\;\;\;J \cdot \left(U \cdot \left(J \cdot U\right)\right)\\ \mathbf{elif}\;\ell \leq 25000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(J \cdot \left(U \cdot U\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -23500.0)
   (* J (* U (* J U)))
   (if (<= l 25000000.0) U (* J (* J (* U U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -23500.0) {
		tmp = J * (U * (J * U));
	} else if (l <= 25000000.0) {
		tmp = U;
	} else {
		tmp = J * (J * (U * U));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-23500.0d0)) then
        tmp = j * (u * (j * u))
    else if (l <= 25000000.0d0) then
        tmp = u
    else
        tmp = j * (j * (u * u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -23500.0) {
		tmp = J * (U * (J * U));
	} else if (l <= 25000000.0) {
		tmp = U;
	} else {
		tmp = J * (J * (U * U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -23500.0:
		tmp = J * (U * (J * U))
	elif l <= 25000000.0:
		tmp = U
	else:
		tmp = J * (J * (U * U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -23500.0)
		tmp = Float64(J * Float64(U * Float64(J * U)));
	elseif (l <= 25000000.0)
		tmp = U;
	else
		tmp = Float64(J * Float64(J * Float64(U * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -23500.0)
		tmp = J * (U * (J * U));
	elseif (l <= 25000000.0)
		tmp = U;
	else
		tmp = J * (J * (U * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -23500.0], N[(J * N[(U * N[(J * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 25000000.0], U, N[(J * N[(J * N[(U * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -23500

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 32.1%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 32.1%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 32.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 32.1%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 32.1%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 32.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 32.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 33.5%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   8. Applied egg-rr 23.8%

      \[\leadsto J \cdot \color{blue}{\left(U \cdot \left(J \cdot U\right)\right)} \]

   if -23500 < l < 2.5e7

   1. Initial program 68.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 64.7%

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

   if 2.5e7 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 33.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 33.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 33.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 33.8%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 33.8%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 33.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 44.9%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   8. Applied egg-rr 23.1%

      \[\leadsto J \cdot \color{blue}{\left(U \cdot \left(J \cdot U\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 23.1%

         \[\leadsto J \cdot \color{blue}{\left(\left(J \cdot U\right) \cdot U\right)} \]

      2. associate-*l* 27.3%

         \[\leadsto J \cdot \color{blue}{\left(J \cdot \left(U \cdot U\right)\right)} \]

   10. Simplified 27.3%

       \[\leadsto J \cdot \color{blue}{\left(J \cdot \left(U \cdot U\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -23500:\\ \;\;\;\;J \cdot \left(U \cdot \left(J \cdot U\right)\right)\\ \mathbf{elif}\;\ell \leq 25000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(J \cdot \left(U \cdot U\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.3% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -39 \lor \neg \left(\ell \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;J \cdot \left(J \cdot U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -39.0) (not (<= l 1.8e+16))) (* J (* J U)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -39.0) || !(l <= 1.8e+16)) {
		tmp = J * (J * U);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-39.0d0)) .or. (.not. (l <= 1.8d+16))) then
        tmp = j * (j * u)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -39.0) || !(l <= 1.8e+16)) {
		tmp = J * (J * U);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -39.0) or not (l <= 1.8e+16):
		tmp = J * (J * U)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -39.0) || !(l <= 1.8e+16))
		tmp = Float64(J * Float64(J * U));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -39.0) || ~((l <= 1.8e+16)))
		tmp = J * (J * U);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -39.0], N[Not[LessEqual[l, 1.8e+16]], $MachinePrecision]], N[(J * N[(J * U), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -39 or 1.8e16 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 32.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 32.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 32.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 32.8%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 32.8%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 32.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 32.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 39.2%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   8. Applied egg-rr 19.6%

      \[\leadsto J \cdot \color{blue}{\left(U \cdot J\right)} \]

   if -39 < l < 1.8e16

   1. Initial program 68.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 65.1%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -39 \lor \neg \left(\ell \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;J \cdot \left(J \cdot U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -23500 \lor \neg \left(\ell \leq 680\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -23500.0) (not (<= l 680.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23500.0) || !(l <= 680.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-23500.0d0)) .or. (.not. (l <= 680.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23500.0) || !(l <= 680.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -23500.0) or not (l <= 680.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -23500.0) || !(l <= 680.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -23500.0) || ~((l <= 680.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -23500.0], N[Not[LessEqual[l, 680.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -23500 or 680 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{U \cdot U} \]

   if -23500 < l < 680

   1. Initial program 68.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 65.2%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -23500 \lor \neg \left(\ell \leq 680\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 39.2% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+36}:\\ \;\;\;\;J \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -2.9e+36) (* J U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.9e+36) {
		tmp = J * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-2.9d+36)) then
        tmp = j * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.9e+36) {
		tmp = J * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.9e+36:
		tmp = J * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.9e+36)
		tmp = Float64(J * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.9e+36)
		tmp = J * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.9e+36], N[(J * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.9e36

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 40.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 40.2%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 40.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

      3. associate-*r* 40.2%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

      4. *-commutative 40.2%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      5. associate-*r* 40.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 40.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 40.2%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   8. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{U}\right)}^{J}\right)} \]
   9. Step-by-step derivation

      1. exp-prod 34.1%

         \[\leadsto \log \color{blue}{\left(e^{U \cdot J}\right)} \]

      2. *-commutative 34.1%

         \[\leadsto \log \left(e^{\color{blue}{J \cdot U}}\right) \]

      3. rem-log-exp 14.6%

         \[\leadsto \color{blue}{J \cdot U} \]

   10. Simplified 14.6%

       \[\leadsto \color{blue}{J \cdot U} \]

   if -2.9e36 < l

   1. Initial program 80.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 40.2%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+36}:\\ \;\;\;\;J \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 55.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + 2 \cdot \left(\ell \cdot J\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* 2.0 (* l J))))

C

double code(double J, double l, double K, double U) {
	return U + (2.0 * (l * J));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (2.0d0 * (l * j))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (2.0 * (l * J));
}

Python

def code(J, l, K, U):
	return U + (2.0 * (l * J))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(2.0 * Float64(l * J)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (2.0 * (l * J));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   2. *-commutative 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

   3. associate-*r* 64.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

   4. *-commutative 64.5%

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

5. Simplified 64.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

6. Taylor expanded in K around 0 53.3%

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]

7. Final simplification 53.3%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]
8. Add Preprocessing

Alternative 23: 2.8% accurate, 312.0× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 1.0;
}

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

4. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{\frac{U}{U}} \]
5. Step-by-step derivation

   1. *-inverses 2.8%

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

6. Simplified 2.8%

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

7. Final simplification 2.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 24: 3.0% accurate, 312.0× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 J)

C

double code(double J, double l, double K, double U) {
	return J;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = j
end function

Java

public static double code(double J, double l, double K, double U) {
	return J;
}

Python

def code(J, l, K, U):
	return J

Julia

function code(J, l, K, U)
	return J
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = J;
end

Wolfram

code[J_, l_, K_, U_] := J

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   2. *-commutative 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)} + U \]

   3. associate-*r* 64.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)} + U \]

   4. *-commutative 64.5%

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. associate-*r* 64.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

5. Simplified 64.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

6. Taylor expanded in J around inf 62.3%

   \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

8. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{\frac{J \cdot U}{U}} \]
9. Step-by-step derivation

   1. associate-/l* 3.1%

      \[\leadsto \color{blue}{J \cdot \frac{U}{U}} \]

   2. *-inverses 3.1%

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

   3. *-rgt-identity 3.1%

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

10. Simplified 3.1%

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

11. Final simplification 3.1%

    \[\leadsto J \]
12. Add Preprocessing

Alternative 25: 37.8% accurate, 312.0× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 84.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0 33.0%

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

4. Final simplification 33.0%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))