Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.9%

Time: 10.7s

Alternatives: 15

Speedup: 1.4×

• 49.2% 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: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -200 \lor \neg \left(t\_1 \leq 0.005\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 -200.0) (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 <= -200.0) || !(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;
}

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 = exp(l) - exp(-l)
    if ((t_1 <= (-200.0d0)) .or. (.not. (t_1 <= 0.005d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * ((0.016666666666666666d0 * (l ** 5.0d0)) + ((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 t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -200.0) || !(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 <= -200.0) 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 <= -200.0) || !(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 <= -200.0) || ~((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, -200.0], 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))) < -200 or 0.0050000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -200 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0050000000000000001

   1. Initial program 67.1%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -200 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -200 \lor \neg \left(t\_1 \leq 0.0005\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 -200.0) (not (<= t_1 0.0005)))
     (+ (* (* 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 <= -200.0) || !(t_1 <= 0.0005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-200.0d0)) .or. (.not. (t_1 <= 0.0005d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (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 t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -200.0) || !(t_1 <= 0.0005)) {
		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 <= -200.0) or not (t_1 <= 0.0005):
		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 <= -200.0) || !(t_1 <= 0.0005))
		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 <= -200.0) || ~((t_1 <= 0.0005)))
		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, -200.0], N[Not[LessEqual[t$95$1, 0.0005]], $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))) < -200 or 5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   if -200 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-4

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -200 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.85:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.0006:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* J (* 0.016666666666666666 (pow l 5.0))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1.1e+68)
     t_0
     (if (<= l -0.85)
       t_1
       (if (<= l 0.0006)
         (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
         (if (<= l 1.35e+56) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (0.016666666666666666 * pow(l, 5.0))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.1e+68) {
		tmp = t_0;
	} else if (l <= -0.85) {
		tmp = t_1;
	} else if (l <= 0.0006) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else if (l <= 1.35e+56) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * (0.016666666666666666d0 * (l ** 5.0d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1.1d+68)) then
        tmp = t_0
    else if (l <= (-0.85d0)) then
        tmp = t_1
    else if (l <= 0.0006d0) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else if (l <= 1.35d+56) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (J * (0.016666666666666666 * Math.pow(l, 5.0))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.1e+68) {
		tmp = t_0;
	} else if (l <= -0.85) {
		tmp = t_1;
	} else if (l <= 0.0006) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else if (l <= 1.35e+56) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (0.016666666666666666 * math.pow(l, 5.0))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1.1e+68:
		tmp = t_0
	elif l <= -0.85:
		tmp = t_1
	elif l <= 0.0006:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	elif l <= 1.35e+56:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(0.016666666666666666 * (l ^ 5.0)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1.1e+68)
		tmp = t_0;
	elseif (l <= -0.85)
		tmp = t_1;
	elseif (l <= 0.0006)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	elseif (l <= 1.35e+56)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (0.016666666666666666 * (l ^ 5.0))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1.1e+68)
		tmp = t_0;
	elseif (l <= -0.85)
		tmp = t_1;
	elseif (l <= 0.0006)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	elseif (l <= 1.35e+56)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.1e+68], t$95$0, If[LessEqual[l, -0.85], t$95$1, If[LessEqual[l, 0.0006], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+56], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.09999999999999994e68 or 1.35000000000000005e56 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -1.09999999999999994e68 < l < -0.849999999999999978 or 5.99999999999999947e-4 < l < 1.35000000000000005e56

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 90.0%

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

   if -0.849999999999999978 < l < 5.99999999999999947e-4

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.85:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.0006:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t\_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.85:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.002:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* J (* 0.016666666666666666 (pow l 5.0))))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1.1e+68)
     t_1
     (if (<= l -0.85)
       t_2
       (if (<= l 0.002)
         (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 1.35e+56) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (J * (0.016666666666666666 * pow(l, 5.0))));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.1e+68) {
		tmp = t_1;
	} else if (l <= -0.85) {
		tmp = t_2;
	} else if (l <= 0.002) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.35e+56) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * (j * (0.016666666666666666d0 * (l ** 5.0d0))))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1.1d+68)) then
        tmp = t_1
    else if (l <= (-0.85d0)) then
        tmp = t_2
    else if (l <= 0.002d0) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 1.35d+56) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (J * (0.016666666666666666 * Math.pow(l, 5.0))));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.1e+68) {
		tmp = t_1;
	} else if (l <= -0.85) {
		tmp = t_2;
	} else if (l <= 0.002) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.35e+56) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (J * (0.016666666666666666 * math.pow(l, 5.0))))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1.1e+68:
		tmp = t_1
	elif l <= -0.85:
		tmp = t_2
	elif l <= 0.002:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 1.35e+56:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64(J * Float64(0.016666666666666666 * (l ^ 5.0)))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1.1e+68)
		tmp = t_1;
	elseif (l <= -0.85)
		tmp = t_2;
	elseif (l <= 0.002)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 1.35e+56)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * (J * (0.016666666666666666 * (l ^ 5.0))));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1.1e+68)
		tmp = t_1;
	elseif (l <= -0.85)
		tmp = t_2;
	elseif (l <= 0.002)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 1.35e+56)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(J * N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.1e+68], t$95$1, If[LessEqual[l, -0.85], t$95$2, If[LessEqual[l, 0.002], 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, 1.35e+56], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.09999999999999994e68 or 1.35000000000000005e56 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -1.09999999999999994e68 < l < -0.849999999999999978 or 2e-3 < l < 1.35000000000000005e56

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 90.0%

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

   if -0.849999999999999978 < l < 2e-3

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.85:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.002:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.85 \lor \neg \left(\ell \leq 0.0005\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.85) (not (<= l 0.0005)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.85) || !(l <= 0.0005)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (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 ((l <= (-0.85d0)) .or. (.not. (l <= 0.0005d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (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 ((l <= -0.85) || !(l <= 0.0005)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.85) or not (l <= 0.0005):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.85) || !(l <= 0.0005))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.85) || ~((l <= 0.0005)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.85], N[Not[LessEqual[l, 0.0005]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.849999999999999978 or 5.0000000000000001e-4 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 82.6%

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

   if -0.849999999999999978 < l < 5.0000000000000001e-4

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.85 \lor \neg \left(\ell \leq 0.0005\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 0.00088:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))
   (if (<= l -1.25e+86)
     t_0
     (if (<= l -3.2e+18)
       (pow U -4.0)
       (if (<= l 0.00088) (+ U (* l (* (cos (* K 0.5)) (* J 2.0)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	double tmp;
	if (l <= -1.25e+86) {
		tmp = t_0;
	} else if (l <= -3.2e+18) {
		tmp = pow(U, -4.0);
	} else if (l <= 0.00088) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    if (l <= (-1.25d+86)) then
        tmp = t_0
    else if (l <= (-3.2d+18)) then
        tmp = u ** (-4.0d0)
    else if (l <= 0.00088d0) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	double tmp;
	if (l <= -1.25e+86) {
		tmp = t_0;
	} else if (l <= -3.2e+18) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 0.00088) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	tmp = 0
	if l <= -1.25e+86:
		tmp = t_0
	elif l <= -3.2e+18:
		tmp = math.pow(U, -4.0)
	elif l <= 0.00088:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))
	tmp = 0.0
	if (l <= -1.25e+86)
		tmp = t_0;
	elseif (l <= -3.2e+18)
		tmp = U ^ -4.0;
	elseif (l <= 0.00088)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	tmp = 0.0;
	if (l <= -1.25e+86)
		tmp = t_0;
	elseif (l <= -3.2e+18)
		tmp = U ^ -4.0;
	elseif (l <= 0.00088)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.25e+86], t$95$0, If[LessEqual[l, -3.2e+18], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 0.00088], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.2499999999999999e86 or 8.80000000000000031e-4 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 81.5%

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

   4. Taylor expanded in K around 0 66.7%

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

   if -1.2499999999999999e86 < l < -3.2e18

   1. Initial program 100.0%

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

   4. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -3.2e18 < l < 8.80000000000000031e-4

   1. Initial program 67.9%

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

   3. Taylor expanded in l around 0 96.7%

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

      1. associate-*r* 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*l* 96.8%

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

   5. Simplified 96.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+86}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 0.00088:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+178} \lor \neg \left(\ell \leq -3 \cdot 10^{+18}\right) \land \ell \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.35e+178) (and (not (<= l -3e+18)) (<= l 1.6e+112)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (pow U -4.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.35e+178) || (!(l <= -3e+18) && (l <= 1.6e+112))) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.35d+178)) .or. (.not. (l <= (-3d+18))) .and. (l <= 1.6d+112)) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.35e+178) || (!(l <= -3e+18) && (l <= 1.6e+112))) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.35e+178) or (not (l <= -3e+18) and (l <= 1.6e+112)):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.35e+178) || (!(l <= -3e+18) && (l <= 1.6e+112)))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.35e+178) || (~((l <= -3e+18)) && (l <= 1.6e+112)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.35e+178], And[N[Not[LessEqual[l, -3e+18]], $MachinePrecision], LessEqual[l, 1.6e+112]]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.35000000000000009e178 or -3e18 < l < 1.59999999999999993e112

   1. Initial program 77.1%

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

   3. Taylor expanded in l around 0 77.4%

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

   if -1.35000000000000009e178 < l < -3e18 or 1.59999999999999993e112 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{{U}^{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+178} \lor \neg \left(\ell \leq -3 \cdot 10^{+18}\right) \land \ell \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+176}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{+18} \lor \neg \left(\ell \leq 5.2 \cdot 10^{+118}\right):\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(t\_0 \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= l -3.1e+176)
     (+ U (* 2.0 (* J (* l t_0))))
     (if (or (<= l -2.8e+18) (not (<= l 5.2e+118)))
       (pow U -4.0)
       (+ U (* l (* t_0 (* J 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (l <= -3.1e+176) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if ((l <= -2.8e+18) || !(l <= 5.2e+118)) {
		tmp = pow(U, -4.0);
	} else {
		tmp = U + (l * (t_0 * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    if (l <= (-3.1d+176)) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else if ((l <= (-2.8d+18)) .or. (.not. (l <= 5.2d+118))) then
        tmp = u ** (-4.0d0)
    else
        tmp = u + (l * (t_0 * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -3.1e+176) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if ((l <= -2.8e+18) || !(l <= 5.2e+118)) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = U + (l * (t_0 * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if l <= -3.1e+176:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif (l <= -2.8e+18) or not (l <= 5.2e+118):
		tmp = math.pow(U, -4.0)
	else:
		tmp = U + (l * (t_0 * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -3.1e+176)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif ((l <= -2.8e+18) || !(l <= 5.2e+118))
		tmp = U ^ -4.0;
	else
		tmp = Float64(U + Float64(l * Float64(t_0 * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -3.1e+176)
		tmp = U + (2.0 * (J * (l * t_0)));
	elseif ((l <= -2.8e+18) || ~((l <= 5.2e+118)))
		tmp = U ^ -4.0;
	else
		tmp = U + (l * (t_0 * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.1e+176], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -2.8e+18], N[Not[LessEqual[l, 5.2e+118]], $MachinePrecision]], N[Power[U, -4.0], $MachinePrecision], N[(U + N[(l * N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.0999999999999999e176

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 40.5%

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

   if -3.0999999999999999e176 < l < -2.8e18 or 5.20000000000000032e118 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -2.8e18 < l < 5.20000000000000032e118

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0 82.9%

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

      1. associate-*r* 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-*l* 82.9%

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

   5. Simplified 82.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+176}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{+18} \lor \neg \left(\ell \leq 5.2 \cdot 10^{+118}\right):\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+178}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{+19} \lor \neg \left(\ell \leq 3.9 \cdot 10^{+114}\right):\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.6e+178)
   (+ U (* 2.0 (* l J)))
   (if (or (<= l -5.8e+19) (not (<= l 3.9e+114)))
     (pow U -4.0)
     (fma (* J 2.0) l U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.6e+178) {
		tmp = U + (2.0 * (l * J));
	} else if ((l <= -5.8e+19) || !(l <= 3.9e+114)) {
		tmp = pow(U, -4.0);
	} else {
		tmp = fma((J * 2.0), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.6e+178)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif ((l <= -5.8e+19) || !(l <= 3.9e+114))
		tmp = U ^ -4.0;
	else
		tmp = fma(Float64(J * 2.0), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.6e+178], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -5.8e+19], N[Not[LessEqual[l, 3.9e+114]], $MachinePrecision]], N[Power[U, -4.0], $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5999999999999998e178

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 40.5%

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

   4. Taylor expanded in K around 0 35.8%

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

   if -3.5999999999999998e178 < l < -5.8e19 or 3.9000000000000001e114 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -5.8e19 < l < 3.9000000000000001e114

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0 82.9%

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

      1. associate-*r* 82.9%

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

      2. fma-define 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   5. Applied egg-rr 82.9%

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

   6. Taylor expanded in K around 0 71.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+178}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{+19} \lor \neg \left(\ell \leq 3.9 \cdot 10^{+114}\right):\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+179} \lor \neg \left(\ell \leq -2.8 \cdot 10^{+18}\right) \land \ell \leq 4.4 \cdot 10^{+113}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1e+179) (and (not (<= l -2.8e+18)) (<= l 4.4e+113)))
   (+ U (* 2.0 (* l J)))
   (pow U -4.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1e+179) || (!(l <= -2.8e+18) && (l <= 4.4e+113))) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1d+179)) .or. (.not. (l <= (-2.8d+18))) .and. (l <= 4.4d+113)) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1e+179) || (!(l <= -2.8e+18) && (l <= 4.4e+113))) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1e+179) or (not (l <= -2.8e+18) and (l <= 4.4e+113)):
		tmp = U + (2.0 * (l * J))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1e+179) || (!(l <= -2.8e+18) && (l <= 4.4e+113)))
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1e+179) || (~((l <= -2.8e+18)) && (l <= 4.4e+113)))
		tmp = U + (2.0 * (l * J));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1e+179], And[N[Not[LessEqual[l, -2.8e+18]], $MachinePrecision], LessEqual[l, 4.4e+113]]], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.9999999999999998e178 or -2.8e18 < l < 4.40000000000000021e113

   1. Initial program 77.1%

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

   3. Taylor expanded in l around 0 77.4%

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

   4. Taylor expanded in K around 0 66.5%

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

   if -9.9999999999999998e178 < l < -2.8e18 or 4.40000000000000021e113 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{{U}^{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+179} \lor \neg \left(\ell \leq -2.8 \cdot 10^{+18}\right) \land \ell \leq 4.4 \cdot 10^{+113}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.2% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 0.00023\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.8e+42) (not (<= l 0.00023))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.8e+42) || !(l <= 0.00023)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.8d+42)) .or. (.not. (l <= 0.00023d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.8e+42) || !(l <= 0.00023)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.8e+42) or not (l <= 0.00023):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.8e+42) || !(l <= 0.00023))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.8e+42) || ~((l <= 0.00023)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.8e+42], N[Not[LessEqual[l, 0.00023]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -4.7999999999999997e42 or 2.3000000000000001e-4 < 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. Add Preprocessing

   4. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -4.7999999999999997e42 < l < 2.3000000000000001e-4

   1. Initial program 68.9%

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

   3. Taylor expanded in J around 0 61.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 0.00023\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.3% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \lor \neg \left(\ell \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.3) (not (<= l 2.6e-8))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.3) || !(l <= 2.6e-8)) {
		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 <= (-5.3d0)) .or. (.not. (l <= 2.6d-8))) 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 <= -5.3) || !(l <= 2.6e-8)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.3) or not (l <= 2.6e-8):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.3) || !(l <= 2.6e-8))
		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 <= -5.3) || ~((l <= 2.6e-8)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.3], N[Not[LessEqual[l, 2.6e-8]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.29999999999999982 or 2.6000000000000001e-8 < l

   1. Initial program 99.6%

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

   4. Applied egg-rr 12.5%

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

   if -5.29999999999999982 < l < 2.6000000000000001e-8

   1. Initial program 66.7%

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

   3. Taylor expanded in J around 0 66.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \lor \neg \left(\ell \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.0% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in l around 0 61.0%

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

4. Taylor expanded in K around 0 52.5%

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

5. Final simplification 52.5%

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

Alternative 14: 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 83.3%

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

4. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

6. Simplified 2.6%

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

7. Final simplification 2.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 15: 36.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in J around 0 34.1%

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

4. Final simplification 34.1%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))