Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.5%

Time: 10.3s

Alternatives: 17

Speedup: 1.5×

• 49.7% 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: 86.0% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-15)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.3333333333333333 (pow l 3.0))
         (+ (* 0.016666666666666666 (pow l 5.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-15)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (l * 2.0)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-15)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + (l * 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-15):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + (l * 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-15))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(l * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-15)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + ((0.016666666666666666 * (l ^ 5.0)) + (l * 2.0)))));
	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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-15]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000002e-15 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000002e-15

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-15)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-15)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-15)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-15):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-15))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-15)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-15]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000002e-15 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000002e-15

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.108:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 5.0) (* J 0.016666666666666666)))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -3e+72)
     t_1
     (if (<= l -0.108)
       t_2
       (if (<= l 1.25e-8)
         (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 8e+50) 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, 5.0) * (J * 0.016666666666666666)));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -3e+72) {
		tmp = t_1;
	} else if (l <= -0.108) {
		tmp = t_2;
	} else if (l <= 1.25e-8) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 8e+50) {
		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 ** 5.0d0) * (j * 0.016666666666666666d0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-3d+72)) then
        tmp = t_1
    else if (l <= (-0.108d0)) then
        tmp = t_2
    else if (l <= 1.25d-8) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 8d+50) 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, 5.0) * (J * 0.016666666666666666)));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -3e+72) {
		tmp = t_1;
	} else if (l <= -0.108) {
		tmp = t_2;
	} else if (l <= 1.25e-8) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 8e+50) {
		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, 5.0) * (J * 0.016666666666666666)))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -3e+72:
		tmp = t_1
	elif l <= -0.108:
		tmp = t_2
	elif l <= 1.25e-8:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 8e+50:
		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 ^ 5.0) * Float64(J * 0.016666666666666666))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -3e+72)
		tmp = t_1;
	elseif (l <= -0.108)
		tmp = t_2;
	elseif (l <= 1.25e-8)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 8e+50)
		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 ^ 5.0) * (J * 0.016666666666666666)));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -3e+72)
		tmp = t_1;
	elseif (l <= -0.108)
		tmp = t_2;
	elseif (l <= 1.25e-8)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 8e+50)
		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, 5.0], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -3e+72], t$95$1, If[LessEqual[l, -0.108], t$95$2, If[LessEqual[l, 1.25e-8], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+50], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.00000000000000003e72 or 8.0000000000000006e50 < 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. Taylor expanded in l around 0 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 99.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 99.0%

         \[\leadsto \color{blue}{\left(\left(0.016666666666666666 \cdot {\ell}^{5}\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 99.0%

         \[\leadsto \left(\color{blue}{\left({\ell}^{5} \cdot 0.016666666666666666\right)} \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 99.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \left(0.016666666666666666 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \left(0.016666666666666666 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -3.00000000000000003e72 < l < -0.107999999999999999 or 1.2499999999999999e-8 < l < 8.0000000000000006e50

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 89.7%

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

   if -0.107999999999999999 < l < 1.2499999999999999e-8

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+72}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -0.108:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.0033:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 5.0) (* J 0.016666666666666666)))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -3e+72)
     t_1
     (if (<= l -0.0033)
       t_2
       (if (<= l 1.25e-8)
         (+ U (* t_0 (* J (* l 2.0))))
         (if (<= l 7.5e+50) 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, 5.0) * (J * 0.016666666666666666)));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -3e+72) {
		tmp = t_1;
	} else if (l <= -0.0033) {
		tmp = t_2;
	} else if (l <= 1.25e-8) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (l <= 7.5e+50) {
		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 ** 5.0d0) * (j * 0.016666666666666666d0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-3d+72)) then
        tmp = t_1
    else if (l <= (-0.0033d0)) then
        tmp = t_2
    else if (l <= 1.25d-8) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else if (l <= 7.5d+50) 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, 5.0) * (J * 0.016666666666666666)));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -3e+72) {
		tmp = t_1;
	} else if (l <= -0.0033) {
		tmp = t_2;
	} else if (l <= 1.25e-8) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (l <= 7.5e+50) {
		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, 5.0) * (J * 0.016666666666666666)))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -3e+72:
		tmp = t_1
	elif l <= -0.0033:
		tmp = t_2
	elif l <= 1.25e-8:
		tmp = U + (t_0 * (J * (l * 2.0)))
	elif l <= 7.5e+50:
		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 ^ 5.0) * Float64(J * 0.016666666666666666))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -3e+72)
		tmp = t_1;
	elseif (l <= -0.0033)
		tmp = t_2;
	elseif (l <= 1.25e-8)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	elseif (l <= 7.5e+50)
		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 ^ 5.0) * (J * 0.016666666666666666)));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -3e+72)
		tmp = t_1;
	elseif (l <= -0.0033)
		tmp = t_2;
	elseif (l <= 1.25e-8)
		tmp = U + (t_0 * (J * (l * 2.0)));
	elseif (l <= 7.5e+50)
		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, 5.0], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -3e+72], t$95$1, If[LessEqual[l, -0.0033], t$95$2, If[LessEqual[l, 1.25e-8], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+50], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.00000000000000003e72 or 7.4999999999999999e50 < 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. Taylor expanded in l around 0 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 99.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 99.0%

         \[\leadsto \color{blue}{\left(\left(0.016666666666666666 \cdot {\ell}^{5}\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 99.0%

         \[\leadsto \left(\color{blue}{\left({\ell}^{5} \cdot 0.016666666666666666\right)} \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 99.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \left(0.016666666666666666 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \left(0.016666666666666666 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -3.00000000000000003e72 < l < -0.0033 or 1.2499999999999999e-8 < l < 7.4999999999999999e50

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 89.7%

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

   if -0.0033 < l < 1.2499999999999999e-8

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+72}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0033:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 5: 87.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0023 \lor \neg \left(\ell \leq 1.25 \cdot 10^{-8}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0023) (not (<= l 1.25e-8)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0023) || !(l <= 1.25e-8)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (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 <= (-0.0023d0)) .or. (.not. (l <= 1.25d-8))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (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 <= -0.0023) || !(l <= 1.25e-8)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.0023) or not (l <= 1.25e-8):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0023) || !(l <= 1.25e-8))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.0023) || ~((l <= 1.25e-8)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0023], N[Not[LessEqual[l, 1.25e-8]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0023 or 1.2499999999999999e-8 < 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. Taylor expanded in K around 0 77.3%

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

   if -0.0023 < l < 1.2499999999999999e-8

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0023 \lor \neg \left(\ell \leq 1.25 \cdot 10^{-8}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 6: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ t_1 := U + 2 \cdot \left(J \cdot \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \ell, \ell\right)\right)\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (log1p (expm1 U)))
        (t_1 (+ U (* 2.0 (* J (fma (* -0.125 (* K K)) l l))))))
   (if (<= l -2.8e+95)
     t_1
     (if (<= l -4.6e+17)
       t_0
       (if (<= l 3400.0)
         (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
         (if (<= l 1.8e+51) (pow U -8.0) (if (<= l 1.95e+238) t_1 t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = log1p(expm1(U));
	double t_1 = U + (2.0 * (J * fma((-0.125 * (K * K)), l, l)));
	double tmp;
	if (l <= -2.8e+95) {
		tmp = t_1;
	} else if (l <= -4.6e+17) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.8e+51) {
		tmp = pow(U, -8.0);
	} else if (l <= 1.95e+238) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = log1p(expm1(U))
	t_1 = Float64(U + Float64(2.0 * Float64(J * fma(Float64(-0.125 * Float64(K * K)), l, l))))
	tmp = 0.0
	if (l <= -2.8e+95)
		tmp = t_1;
	elseif (l <= -4.6e+17)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.8e+51)
		tmp = U ^ -8.0;
	elseif (l <= 1.95e+238)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(J * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] * l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.8e+95], t$95$1, If[LessEqual[l, -4.6e+17], t$95$0, If[LessEqual[l, 3400.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+51], N[Power[U, -8.0], $MachinePrecision], If[LessEqual[l, 1.95e+238], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.7999999999999998e95 or 1.80000000000000005e51 < l < 1.94999999999999996e238

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 98.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 32.9%

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

      1. associate-*r* 32.9%

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

   5. Simplified 32.9%

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

   6. Taylor expanded in K around 0 41.2%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(-0.125 \cdot \left({K}^{2} \cdot \ell\right) + \ell\right)} \cdot J\right) + U \]
   7. Step-by-step derivation

      1. associate-*r* 41.2%

         \[\leadsto 2 \cdot \left(\left(\color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \ell} + \ell\right) \cdot J\right) + U \]

      2. fma-def 41.2%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.125 \cdot {K}^{2}, \ell, \ell\right)} \cdot J\right) + U \]

      3. unpow2 41.2%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)}, \ell, \ell\right) \cdot J\right) + U \]

   8. Simplified 41.2%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \ell, \ell\right)} \cdot J\right) + U \]

   if -2.7999999999999998e95 < l < -4.6e17 or 1.94999999999999996e238 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)} \]

   if -4.6e17 < l < 3400

   1. Initial program 70.6%

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

   2. Taylor expanded in l around 0 97.1%

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

   if 3400 < l < 1.80000000000000005e51

   1. Initial program 100.0%

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

   3. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \ell, \ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+238}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{fma}\left(-0.125 \cdot \left(K \cdot K\right), \ell, \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \end{array} \]

Alternative 7: 63.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in l around 0 88.4%

   \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

3. Taylor expanded in l around 0 62.1%

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

   1. associate-*r* 62.0%

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

5. Simplified 62.0%

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

6. Final simplification 62.0%

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

Alternative 8: 63.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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 + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in l around 0 85.3%

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

3. Taylor expanded in l around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-*r* 62.0%

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

   3. *-commutative 62.0%

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

   4. *-commutative 62.0%

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

   5. associate-*r* 62.0%

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

   6. associate-*l* 62.0%

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

   7. *-commutative 62.0%

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

   8. *-commutative 62.0%

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

5. Simplified 62.0%

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

6. Final simplification 62.0%

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

Alternative 9: 63.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (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 + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in l around 0 62.1%

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

3. Final simplification 62.1%

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

Alternative 10: 59.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+55}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -1.4e+47)
     t_0
     (if (<= l 3400.0)
       (fma (* l 2.0) J U)
       (if (<= l 1.8e+55) (pow U -8.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -1.4e+47) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = fma((l * 2.0), J, U);
	} else if (l <= 1.8e+55) {
		tmp = pow(U, -8.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -1.4e+47)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = fma(Float64(l * 2.0), J, U);
	elseif (l <= 1.8e+55)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+47], t$95$0, If[LessEqual[l, 3400.0], N[(N[(l * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 1.8e+55], N[Power[U, -8.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.39999999999999994e47 or 1.79999999999999994e55 < 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. Taylor expanded in l around 0 94.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 31.5%

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

      1. associate-*r* 31.5%

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

   5. Simplified 31.5%

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

   6. Taylor expanded in K around 0 15.9%

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

      1. +-commutative 15.9%

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

      2. associate-*r* 15.9%

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

      3. distribute-rgt-out 35.9%

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

      4. *-commutative 35.9%

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

      5. unpow2 35.9%

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

   8. Simplified 35.9%

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

   if -1.39999999999999994e47 < l < 3400

   1. Initial program 73.7%

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

   2. Taylor expanded in l around 0 89.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 88.0%

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

      1. associate-*r* 87.9%

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

   5. Simplified 87.9%

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

   6. Taylor expanded in K around 0 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

      3. fma-def 75.0%

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

   8. Simplified 75.0%

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

   if 3400 < l < 1.79999999999999994e55

   1. Initial program 100.0%

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

   3. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+55}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 11: 59.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 10^{+51}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -1.2e+47)
     t_0
     (if (<= l 3400.0)
       (+ U (* 2.0 (* l J)))
       (if (<= l 1e+51) (pow U -8.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -1.2e+47) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1e+51) {
		tmp = pow(U, -8.0);
	} 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) :: tmp
    t_0 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (l <= (-1.2d+47)) then
        tmp = t_0
    else if (l <= 3400.0d0) then
        tmp = u + (2.0d0 * (l * j))
    else if (l <= 1d+51) then
        tmp = u ** (-8.0d0)
    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 + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -1.2e+47) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1e+51) {
		tmp = Math.pow(U, -8.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if l <= -1.2e+47:
		tmp = t_0
	elif l <= 3400.0:
		tmp = U + (2.0 * (l * J))
	elif l <= 1e+51:
		tmp = math.pow(U, -8.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -1.2e+47)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif (l <= 1e+51)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (l <= -1.2e+47)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = U + (2.0 * (l * J));
	elseif (l <= 1e+51)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.2e+47], t$95$0, If[LessEqual[l, 3400.0], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+51], N[Power[U, -8.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.20000000000000009e47 or 1e51 < 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. Taylor expanded in l around 0 94.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 31.5%

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

      1. associate-*r* 31.5%

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

   5. Simplified 31.5%

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

   6. Taylor expanded in K around 0 15.9%

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

      1. +-commutative 15.9%

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

      2. associate-*r* 15.9%

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

      3. distribute-rgt-out 35.9%

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

      4. *-commutative 35.9%

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

      5. unpow2 35.9%

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

   8. Simplified 35.9%

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

   if -1.20000000000000009e47 < l < 3400

   1. Initial program 73.7%

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

   2. Taylor expanded in l around 0 89.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 88.0%

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

      1. associate-*r* 87.9%

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

   5. Simplified 87.9%

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

   6. Taylor expanded in K around 0 75.0%

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

   if 3400 < l < 1e51

   1. Initial program 100.0%

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

   3. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 10^{+51}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 12: 59.2% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+47} \lor \neg \left(\ell \leq 10^{+20}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.2e+47) (not (<= l 1e+20)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e+47) || !(l <= 1e+20)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} 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 ((l <= (-1.2d+47)) .or. (.not. (l <= 1d+20))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    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 ((l <= -1.2e+47) || !(l <= 1e+20)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.2e+47) or not (l <= 1e+20):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.2e+47) || !(l <= 1e+20))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	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 ((l <= -1.2e+47) || ~((l <= 1e+20)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.2e+47], N[Not[LessEqual[l, 1e+20]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.20000000000000009e47 or 1e20 < 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. Taylor expanded in l around 0 90.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 30.6%

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

      1. associate-*r* 30.6%

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

   5. Simplified 30.6%

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

   6. Taylor expanded in K around 0 15.0%

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

      1. +-commutative 15.0%

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

      2. associate-*r* 15.0%

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

      3. distribute-rgt-out 34.7%

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

      4. *-commutative 34.7%

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

      5. unpow2 34.7%

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

   8. Simplified 34.7%

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

   if -1.20000000000000009e47 < l < 1e20

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 87.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 84.7%

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

      1. associate-*r* 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in K around 0 72.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+47} \lor \neg \left(\ell \leq 10^{+20}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 13: 39.2% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00046:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -0.00046) (* U (- U -8.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.00046) {
		tmp = U * (U - -8.0);
	} 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 <= (-0.00046d0)) then
        tmp = u * (u - (-8.0d0))
    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 <= -0.00046) {
		tmp = U * (U - -8.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.00046:
		tmp = U * (U - -8.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.00046)
		tmp = Float64(U * Float64(U - -8.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -0.00046)
		tmp = U * (U - -8.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.00046], N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -4.6000000000000001e-4

   1. Initial program 99.9%

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

   3. Applied egg-rr 14.6%

      \[\leadsto \color{blue}{U \cdot \left(U - -8\right)} \]

   if -4.6000000000000001e-4 < l

   1. Initial program 79.1%

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

   2. Taylor expanded in J around 0 48.6%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00046:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 14: 53.9% 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 85.3%

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

2. Taylor expanded in l around 0 88.4%

   \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

3. Taylor expanded in l around 0 62.1%

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

   1. associate-*r* 62.0%

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

5. Simplified 62.0%

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

6. Taylor expanded in K around 0 51.0%

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

7. Final simplification 51.0%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]

Alternative 15: 39.2% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00046:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -0.00046) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.00046) {
		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 <= (-0.00046d0)) 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 <= -0.00046) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.00046:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.00046)
		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 <= -0.00046)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.00046], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -4.6000000000000001e-4

   1. Initial program 99.9%

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

   3. Applied egg-rr 14.6%

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

   if -4.6000000000000001e-4 < l

   1. Initial program 79.1%

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

   2. Taylor expanded in J around 0 48.6%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00046:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 16: 2.7% 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 85.3%

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

3. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

5. Simplified 2.6%

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

6. Final simplification 2.6%

   \[\leadsto 1 \]

Alternative 17: 36.3% 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 85.3%

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

2. Taylor expanded in J around 0 34.8%

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

3. Final simplification 34.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023189 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))