Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.8%

Time: 14.5s

Alternatives: 18

Speedup: 1.5×

• 49.9% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.005\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 0.005)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 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 <= 0.005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((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 <= 0.005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((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 <= 0.005):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((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 <= 0.005))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + 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 <= 0.005)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((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, 0.005]], $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[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $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 0.0050000000000000001 < (-.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))) < 0.0050000000000000001

   1. Initial program 76.7%

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

   2. Taylor expanded in l around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 0.0)))
     (+ (* (* 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 <= 0.0)) {
		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 <= 0.0)) {
		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 <= 0.0):
		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 <= 0.0))
		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 <= 0.0)))
		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, 0.0]], $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 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 75.9%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 5.0) (* J 0.016666666666666666))))))
   (if (<= l -5500000000000.0)
     t_0
     (if (<= l 3.6e-20)
       (+
        U
        (*
         J
         (* (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)) (cos (* K 0.5)))))
       (if (<= l 3.6e+38) (+ (* (- (exp l) (exp (- l))) J) U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 5.0) * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5500000000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e-20) {
		tmp = U + (J * (((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)) * cos((K * 0.5))));
	} else if (l <= 3.6e+38) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} 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 + (cos((k / 2.0d0)) * ((l ** 5.0d0) * (j * 0.016666666666666666d0)))
    if (l <= (-5500000000000.0d0)) then
        tmp = t_0
    else if (l <= 3.6d-20) then
        tmp = u + (j * (((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)) * cos((k * 0.5d0))))
    else if (l <= 3.6d+38) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    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, 5.0) * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5500000000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e-20) {
		tmp = U + (J * (((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)) * Math.cos((K * 0.5))));
	} else if (l <= 3.6e+38) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 5.0) * (J * 0.016666666666666666)))
	tmp = 0
	if l <= -5500000000000.0:
		tmp = t_0
	elif l <= 3.6e-20:
		tmp = U + (J * (((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)) * math.cos((K * 0.5))))
	elif l <= 3.6e+38:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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 ^ 5.0) * Float64(J * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -5500000000000.0)
		tmp = t_0;
	elseif (l <= 3.6e-20)
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)) * cos(Float64(K * 0.5)))));
	elseif (l <= 3.6e+38)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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 ^ 5.0) * (J * 0.016666666666666666)));
	tmp = 0.0;
	if (l <= -5500000000000.0)
		tmp = t_0;
	elseif (l <= 3.6e-20)
		tmp = U + (J * (((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)) * cos((K * 0.5))));
	elseif (l <= 3.6e+38)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	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, 5.0], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5500000000000.0], t$95$0, If[LessEqual[l, 3.6e-20], N[(U + N[(J * N[(N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e+38], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5e12 or 3.59999999999999969e38 < 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 93.5%

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

   3. Taylor expanded in l around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. *-commutative 93.5%

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

      3. *-commutative 93.5%

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

   5. Simplified 93.5%

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

   if -5.5e12 < l < 3.59999999999999974e-20

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 99.0%

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

   3. Taylor expanded in J around 0 99.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} + U \]

   if 3.59999999999999974e-20 < l < 3.59999999999999969e38

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0 85.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))))))
   (if (<= l -5500000000000.0)
     t_1
     (if (<= l 3.6e-20)
       (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
       (if (<= l 3.6e+38) (+ (* (- (exp l) (exp (- l))) J) U) 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 tmp;
	if (l <= -5500000000000.0) {
		tmp = t_1;
	} else if (l <= 3.6e-20) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 3.6e+38) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} 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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 5.0d0) * (j * 0.016666666666666666d0)))
    if (l <= (-5500000000000.0d0)) then
        tmp = t_1
    else if (l <= 3.6d-20) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 3.6d+38) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    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 tmp;
	if (l <= -5500000000000.0) {
		tmp = t_1;
	} else if (l <= 3.6e-20) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 3.6e+38) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} 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)))
	tmp = 0
	if l <= -5500000000000.0:
		tmp = t_1
	elif l <= 3.6e-20:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 3.6e+38:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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))))
	tmp = 0.0
	if (l <= -5500000000000.0)
		tmp = t_1;
	elseif (l <= 3.6e-20)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 3.6e+38)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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)));
	tmp = 0.0;
	if (l <= -5500000000000.0)
		tmp = t_1;
	elseif (l <= 3.6e-20)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 3.6e+38)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	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]}, If[LessEqual[l, -5500000000000.0], t$95$1, If[LessEqual[l, 3.6e-20], 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, 3.6e+38], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5e12 or 3.59999999999999969e38 < 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 93.5%

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

   3. Taylor expanded in l around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. *-commutative 93.5%

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

      3. *-commutative 93.5%

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

   5. Simplified 93.5%

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

   if -5.5e12 < l < 3.59999999999999974e-20

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 99.0%

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

   if 3.59999999999999974e-20 < l < 3.59999999999999969e38

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0 85.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 5: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 5.0) (* J 0.016666666666666666))))))
   (if (<= l -5500000000000.0)
     t_0
     (if (<= l 3.6e-20)
       (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))
       (if (<= l 3.6e+38) (+ (* (- (exp l) (exp (- l))) J) U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 5.0) * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5500000000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e-20) {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	} else if (l <= 3.6e+38) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} 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 + (cos((k / 2.0d0)) * ((l ** 5.0d0) * (j * 0.016666666666666666d0)))
    if (l <= (-5500000000000.0d0)) then
        tmp = t_0
    else if (l <= 3.6d-20) then
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    else if (l <= 3.6d+38) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    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, 5.0) * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5500000000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e-20) {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	} else if (l <= 3.6e+38) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 5.0) * (J * 0.016666666666666666)))
	tmp = 0
	if l <= -5500000000000.0:
		tmp = t_0
	elif l <= 3.6e-20:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	elif l <= 3.6e+38:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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 ^ 5.0) * Float64(J * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -5500000000000.0)
		tmp = t_0;
	elseif (l <= 3.6e-20)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	elseif (l <= 3.6e+38)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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 ^ 5.0) * (J * 0.016666666666666666)));
	tmp = 0.0;
	if (l <= -5500000000000.0)
		tmp = t_0;
	elseif (l <= 3.6e-20)
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	elseif (l <= 3.6e+38)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	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, 5.0], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5500000000000.0], t$95$0, If[LessEqual[l, 3.6e-20], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e+38], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5e12 or 3.59999999999999969e38 < 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 93.5%

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

   3. Taylor expanded in l around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. *-commutative 93.5%

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

      3. *-commutative 93.5%

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

   5. Simplified 93.5%

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

   if -5.5e12 < l < 3.59999999999999974e-20

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. associate-*l* 98.7%

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

      4. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   if 3.59999999999999974e-20 < l < 3.59999999999999969e38

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0 85.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5500000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 6: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.64)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.64) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (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)) <= (-0.64d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (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)) <= -0.64) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.64:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.64)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + 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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.64)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.64], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.640000000000000013

   1. Initial program 91.1%

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

   2. Taylor expanded in l around 0 45.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 43.1%

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

      4. *-commutative 43.1%

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

      5. associate-*l* 45.0%

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

   4. Simplified 45.0%

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

   5. Taylor expanded in K around 0 57.6%

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

   6. Taylor expanded in K around inf 65.0%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

   8. Simplified 65.0%

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

   if -0.640000000000000013 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.3%

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

   2. Taylor expanded in l around 0 88.7%

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

   3. Taylor expanded in K around 0 81.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.65)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	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)) <= (-0.65d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    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)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.65:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.65)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.65)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.65], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.650000000000000022

   1. Initial program 90.7%

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

   2. Taylor expanded in l around 0 42.8%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r* 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 42.8%

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

   4. Simplified 42.8%

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

   5. Taylor expanded in K around 0 55.9%

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

   6. Taylor expanded in K around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -0.650000000000000022 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 8: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.65)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* (* J 2.0) (* l (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + ((J * 2.0) * (l * cos((K * 0.5))));
	}
	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)) <= (-0.65d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + ((j * 2.0d0) * (l * cos((k * 0.5d0))))
    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)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + ((J * 2.0) * (l * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.65:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + ((J * 2.0) * (l * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.65)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(Float64(J * 2.0) * Float64(l * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.65)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + ((J * 2.0) * (l * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.65], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.650000000000000022

   1. Initial program 90.7%

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

   2. Taylor expanded in l around 0 42.8%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r* 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 42.8%

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

   4. Simplified 42.8%

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

   5. Taylor expanded in K around 0 55.9%

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

   6. Taylor expanded in K around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -0.650000000000000022 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 94.0%

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

   3. Taylor expanded in l around 0 63.7%

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

      1. associate-*r* 63.7%

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

   5. Simplified 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.65)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	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)) <= (-0.65d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    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)) <= -0.65) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.65:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.65)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.65)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.65], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.650000000000000022

   1. Initial program 90.7%

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

   2. Taylor expanded in l around 0 42.8%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r* 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 42.8%

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

   4. Simplified 42.8%

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

   5. Taylor expanded in K around 0 55.9%

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

   6. Taylor expanded in K around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -0.650000000000000022 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*r* 63.7%

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

      3. associate-*l* 63.7%

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

      4. *-commutative 63.7%

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

   4. Simplified 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.65)
     (+ U (* (* J -0.25) (* l (pow K 2.0))))
     (+ U (* t_0 (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.65) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (t_0 * (l * (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((k / 2.0d0))
    if (t_0 <= (-0.65d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (t_0 * (l * (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((K / 2.0));
	double tmp;
	if (t_0 <= -0.65) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (t_0 * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.65:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (t_0 * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.65)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.65)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (t_0 * (l * (J * 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]}, If[LessEqual[t$95$0, -0.65], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.650000000000000022

   1. Initial program 90.7%

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

   2. Taylor expanded in l around 0 42.8%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r* 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 42.8%

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

   4. Simplified 42.8%

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

   5. Taylor expanded in K around 0 55.9%

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

   6. Taylor expanded in K around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -0.650000000000000022 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 94.0%

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

   3. Taylor expanded in l around 0 63.7%

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

      1. associate-*r* 63.7%

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

      2. *-commutative 63.7%

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

   5. Simplified 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

Alternative 11: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + -0.25 \cdot \left(\ell \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.64)
   (+ U (* -0.25 (* l (* J (pow K 2.0)))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.64) {
		tmp = U + (-0.25 * (l * (J * pow(K, 2.0))));
	} else {
		tmp = U + (l * (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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.64d0)) then
        tmp = u + ((-0.25d0) * (l * (j * (k ** 2.0d0))))
    else
        tmp = u + (l * (j * 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)) <= -0.64) {
		tmp = U + (-0.25 * (l * (J * Math.pow(K, 2.0))));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.64:
		tmp = U + (-0.25 * (l * (J * math.pow(K, 2.0))))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.64)
		tmp = Float64(U + Float64(-0.25 * Float64(l * Float64(J * (K ^ 2.0)))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.64)
		tmp = U + (-0.25 * (l * (J * (K ^ 2.0))));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.64], N[(U + N[(-0.25 * N[(l * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.640000000000000013

   1. Initial program 91.1%

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

   2. Taylor expanded in l around 0 45.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 43.1%

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

      4. *-commutative 43.1%

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

      5. associate-*l* 45.0%

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

   4. Simplified 45.0%

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

   5. Taylor expanded in K around 0 57.6%

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

   6. Taylor expanded in K around inf 65.0%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   8. Simplified 57.6%

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

   if -0.640000000000000013 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.3%

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

   2. Taylor expanded in l around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-*l* 63.4%

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

   4. Simplified 63.4%

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

   5. Taylor expanded in K around 0 58.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + -0.25 \cdot \left(\ell \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 12: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.64)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.64) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (l * (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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.64d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (l * (j * 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)) <= -0.64) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.64:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.64)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.64)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.64], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.640000000000000013

   1. Initial program 91.1%

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

   2. Taylor expanded in l around 0 45.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 43.1%

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

      4. *-commutative 43.1%

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

      5. associate-*l* 45.0%

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

   4. Simplified 45.0%

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

   5. Taylor expanded in K around 0 57.6%

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

   6. Taylor expanded in K around inf 65.0%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

   8. Simplified 65.0%

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

   if -0.640000000000000013 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.3%

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

   2. Taylor expanded in l around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-*l* 63.4%

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

   4. Simplified 63.4%

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

   5. Taylor expanded in K around 0 58.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.64:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 13: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.007 \lor \neg \left(\ell \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.007) (not (<= l 3.6e-20)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.007) || !(l <= 3.6e-20)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	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 <= (-0.007d0)) .or. (.not. (l <= 3.6d-20))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.007) || !(l <= 3.6e-20)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.007) or not (l <= 3.6e-20):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.007) || !(l <= 3.6e-20))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.007) || ~((l <= 3.6e-20)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.007], N[Not[LessEqual[l, 3.6e-20]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.00700000000000000015 or 3.59999999999999974e-20 < l

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 70.9%

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

   if -0.00700000000000000015 < l < 3.59999999999999974e-20

   1. Initial program 75.7%

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

   2. Taylor expanded in l around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.6%

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

      3. associate-*l* 99.6%

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

      4. *-commutative 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.007 \lor \neg \left(\ell \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 14: 52.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 4e+226)
   (+ U (* l (* J 2.0)))
   (+ U (* (pow K 2.0) (* J 0.0625)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 4e+226) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (pow(K, 2.0) * (J * 0.0625));
	}
	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 ((k / 2.0d0) <= 4d+226) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + ((k ** 2.0d0) * (j * 0.0625d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 4e+226) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (Math.pow(K, 2.0) * (J * 0.0625));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 4e+226:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (math.pow(K, 2.0) * (J * 0.0625))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 4e+226)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64((K ^ 2.0) * Float64(J * 0.0625)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 4e+226)
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + ((K ^ 2.0) * (J * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 4e+226], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Power[K, 2.0], $MachinePrecision] * N[(J * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K 2) < 3.99999999999999985e226

   1. Initial program 89.7%

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

   2. Taylor expanded in l around 0 61.0%

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

      1. associate-*r* 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*r* 60.5%

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

      4. *-commutative 60.5%

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

      5. associate-*l* 61.0%

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

   4. Simplified 61.0%

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

   5. Taylor expanded in K around 0 52.9%

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

   if 3.99999999999999985e226 < (/.f64 K 2)

   1. Initial program 89.8%

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

   3. Applied egg-rr 13.5%

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

   4. Taylor expanded in K around 0 45.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]

   5. Taylor expanded in K around inf 45.0%

      \[\leadsto \color{blue}{0.0625 \cdot \left(J \cdot {K}^{2}\right)} + U \]
   6. Step-by-step derivation

      1. associate-*r* 45.0%

         \[\leadsto \color{blue}{\left(0.0625 \cdot J\right) \cdot {K}^{2}} + U \]

      2. *-commutative 45.0%

         \[\leadsto \color{blue}{\left(J \cdot 0.0625\right)} \cdot {K}^{2} + U \]

      3. *-commutative 45.0%

         \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot 0.0625\right)} + U \]

   7. Simplified 45.0%

      \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot 0.0625\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot 0.0625\right)\\ \end{array} \]

Alternative 15: 42.1% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+20} \lor \neg \left(\ell \leq 6 \cdot 10^{+30}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.5e+20) (not (<= l 6e+30))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.5e+20) || !(l <= 6e+30)) {
		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 <= (-1.5d+20)) .or. (.not. (l <= 6d+30))) 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 <= -1.5e+20) || !(l <= 6e+30)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.5e+20) or not (l <= 6e+30):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.5e+20) || !(l <= 6e+30))
		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 <= -1.5e+20) || ~((l <= 6e+30)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.5e+20], N[Not[LessEqual[l, 6e+30]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.5e20 or 5.99999999999999956e30 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 15.6%

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

   if -1.5e20 < l < 5.99999999999999956e30

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. fma-def 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in J around 0 68.3%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+20} \lor \neg \left(\ell \leq 6 \cdot 10^{+30}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 16: 53.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in l around 0 59.6%

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

   1. associate-*r* 59.3%

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

   2. *-commutative 59.3%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. associate-*l* 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in K around 0 50.9%

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

6. Final simplification 50.9%

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

Alternative 17: 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 89.7%

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

   1. associate-*l* 89.7%

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

   2. fma-def 89.7%

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

3. Simplified 89.7%

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

5. Applied egg-rr 2.5%

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

   1. *-inverses 2.5%

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

7. Simplified 2.5%

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

8. Final simplification 2.5%

   \[\leadsto 1 \]

Alternative 18: 36.7% 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 89.7%

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

   1. associate-*l* 89.7%

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

   2. fma-def 89.7%

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

3. Simplified 89.7%

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

4. Taylor expanded in J around 0 34.2%

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

5. Final simplification 34.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023301 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))