Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.5%

Time: 15.4s

Alternatives: 16

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* J (log1p (expm1 (* 2.0 (* l (cos (* K 0.5))))))) U))

C

double code(double J, double l, double K, double U) {
	return (J * log1p(expm1((2.0 * (l * cos((K * 0.5))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (J * Math.log1p(Math.expm1((2.0 * (l * Math.cos((K * 0.5))))))) + U;
}

Python

def code(J, l, K, U):
	return (J * math.log1p(math.expm1((2.0 * (l * math.cos((K * 0.5))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(J * log1p(expm1(Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(J * N[Log[1 + N[(Exp[N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Taylor expanded in l around 0 63.1%

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

   1. associate-*r* 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in J around 0 63.1%

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

   1. *-commutative 63.1%

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

   2. associate-*r* 63.1%

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

   3. *-commutative 63.1%

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

   4. associate-*l* 63.1%

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

   5. *-commutative 63.1%

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

8. Simplified 63.1%

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

   1. log1p-expm1-u 99.5%

      \[\leadsto J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell\right)\right)\right)} + U \]

   2. *-commutative 99.5%

      \[\leadsto J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)}\right)\right) + U \]

   3. associate-*l* 99.5%

      \[\leadsto J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}\right)\right) + U \]

   4. *-commutative 99.5%

      \[\leadsto J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

10. Applied egg-rr 99.5%

    \[\leadsto J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)} + U \]

11. Final simplification 99.5%

    \[\leadsto J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \]
12. Add Preprocessing

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.5 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;U + J \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.5) (not (<= t_0 2e-13)))
     (+ U (* J t_0))
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 2e-13)) {
		tmp = U + (J * t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-0.5d0)) .or. (.not. (t_0 <= 2d-13))) then
        tmp = u + (j * t_0)
    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 t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 2e-13)) {
		tmp = U + (J * t_0);
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.5) or not (t_0 <= 2e-13):
		tmp = U + (J * t_0)
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.5) || !(t_0 <= 2e-13))
		tmp = Float64(U + Float64(J * t_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)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -0.5) || ~((t_0 <= 2e-13)))
		tmp = U + (J * t_0);
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.5], N[Not[LessEqual[t$95$0, 2e-13]], $MachinePrecision]], N[(U + N[(J * t$95$0), $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 (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.5 or 2.0000000000000001e-13 < (-.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

   3. Taylor expanded in K around 0 73.4%

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

   if -0.5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-13

   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%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.5 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.086:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1650:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (pow l 7.0) (* J 0.0003968253968253968)) (cos (* K 0.5)))))
   (if (<= l -8e+106)
     t_0
     (if (<= l -0.086)
       (+ U (* J (- (exp l) (exp (- l)))))
       (if (<= l 1650.0)
         (+
          U
          (*
           (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
           (cos (/ K 2.0))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (pow(l, 7.0) * (J * 0.0003968253968253968)) * cos((K * 0.5));
	double tmp;
	if (l <= -8e+106) {
		tmp = t_0;
	} else if (l <= -0.086) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else if (l <= 1650.0) {
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) * cos((K / 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 = ((l ** 7.0d0) * (j * 0.0003968253968253968d0)) * cos((k * 0.5d0))
    if (l <= (-8d+106)) then
        tmp = t_0
    else if (l <= (-0.086d0)) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else if (l <= 1650.0d0) then
        tmp = u + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) * cos((k / 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 = (Math.pow(l, 7.0) * (J * 0.0003968253968253968)) * Math.cos((K * 0.5));
	double tmp;
	if (l <= -8e+106) {
		tmp = t_0;
	} else if (l <= -0.086) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else if (l <= 1650.0) {
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) * Math.cos((K / 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.pow(l, 7.0) * (J * 0.0003968253968253968)) * math.cos((K * 0.5))
	tmp = 0
	if l <= -8e+106:
		tmp = t_0
	elif l <= -0.086:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	elif l <= 1650.0:
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) * math.cos((K / 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968)) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (l <= -8e+106)
		tmp = t_0;
	elseif (l <= -0.086)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	elseif (l <= 1650.0)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) * cos(Float64(K / 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((l ^ 7.0) * (J * 0.0003968253968253968)) * cos((K * 0.5));
	tmp = 0.0;
	if (l <= -8e+106)
		tmp = t_0;
	elseif (l <= -0.086)
		tmp = U + (J * (exp(l) - exp(-l)));
	elseif (l <= 1650.0)
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) * cos((K / 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8e+106], t$95$0, If[LessEqual[l, -0.086], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1650.0], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.00000000000000073e106 or 1650 < 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-define 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%

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

   5. Taylor expanded in l around 0 94.1%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 94.1%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) + \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]

      2. fma-define 94.1%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)}, U\right) \]

      3. *-commutative 94.1%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) \cdot \ell}\right), U\right) \]

      4. *-commutative 94.1%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell\right), U\right) \]

      5. associate-*l* 94.1%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)}\right), U\right) \]

   7. Simplified 94.1%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right)\right) \cdot {\ell}^{3}\right)}, U\right) \]

   8. Taylor expanded in l around inf 94.1%

      \[\leadsto \color{blue}{0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 94.1%

         \[\leadsto 0.0003968253968253968 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{7}\right)}\right) \]

      2. associate-*r* 94.1%

         \[\leadsto 0.0003968253968253968 \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot {\ell}^{7}\right)} \]

      3. associate-*l* 94.1%

         \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{7}} \]

      4. *-commutative 94.1%

         \[\leadsto \color{blue}{{\ell}^{7} \cdot \left(0.0003968253968253968 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

      5. associate-*r* 94.1%

         \[\leadsto {\ell}^{7} \cdot \color{blue}{\left(\left(0.0003968253968253968 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]

      6. associate-*r* 94.1%

         \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(0.0003968253968253968 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      7. *-commutative 94.1%

         \[\leadsto \left({\ell}^{7} \cdot \color{blue}{\left(J \cdot 0.0003968253968253968\right)}\right) \cdot \cos \left(0.5 \cdot K\right) \]

   10. Simplified 94.1%

       \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

   if -8.00000000000000073e106 < l < -0.085999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 88.5%

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

   if -0.085999999999999993 < l < 1650

   1. Initial program 67.8%

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

   3. Taylor expanded in l around 0 99.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+106}:\\ \;\;\;\;\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -0.086:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1650:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot t\_1\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -0.00185:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.00125:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_1\right)\right)\\ \mathbf{elif}\;\ell \leq 170000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (- (exp l) (exp (- l))))))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* (pow l 7.0) (* J 0.0003968253968253968)) t_1)))
   (if (<= l -8e+106)
     t_2
     (if (<= l -0.00185)
       t_0
       (if (<= l 0.00125)
         (+ U (* 2.0 (* J (* l t_1))))
         (if (<= l 170000000000.0) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (exp(l) - exp(-l)));
	double t_1 = cos((K * 0.5));
	double t_2 = (pow(l, 7.0) * (J * 0.0003968253968253968)) * t_1;
	double tmp;
	if (l <= -8e+106) {
		tmp = t_2;
	} else if (l <= -0.00185) {
		tmp = t_0;
	} else if (l <= 0.00125) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 170000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = u + (j * (exp(l) - exp(-l)))
    t_1 = cos((k * 0.5d0))
    t_2 = ((l ** 7.0d0) * (j * 0.0003968253968253968d0)) * t_1
    if (l <= (-8d+106)) then
        tmp = t_2
    else if (l <= (-0.00185d0)) then
        tmp = t_0
    else if (l <= 0.00125d0) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 170000000000.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double t_1 = Math.cos((K * 0.5));
	double t_2 = (Math.pow(l, 7.0) * (J * 0.0003968253968253968)) * t_1;
	double tmp;
	if (l <= -8e+106) {
		tmp = t_2;
	} else if (l <= -0.00185) {
		tmp = t_0;
	} else if (l <= 0.00125) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 170000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (math.exp(l) - math.exp(-l)))
	t_1 = math.cos((K * 0.5))
	t_2 = (math.pow(l, 7.0) * (J * 0.0003968253968253968)) * t_1
	tmp = 0
	if l <= -8e+106:
		tmp = t_2
	elif l <= -0.00185:
		tmp = t_0
	elif l <= 0.00125:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 170000000000.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968)) * t_1)
	tmp = 0.0
	if (l <= -8e+106)
		tmp = t_2;
	elseif (l <= -0.00185)
		tmp = t_0;
	elseif (l <= 0.00125)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 170000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (exp(l) - exp(-l)));
	t_1 = cos((K * 0.5));
	t_2 = ((l ^ 7.0) * (J * 0.0003968253968253968)) * t_1;
	tmp = 0.0;
	if (l <= -8e+106)
		tmp = t_2;
	elseif (l <= -0.00185)
		tmp = t_0;
	elseif (l <= 0.00125)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 170000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[l, -8e+106], t$95$2, If[LessEqual[l, -0.00185], t$95$0, If[LessEqual[l, 0.00125], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 170000000000.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.00000000000000073e106 or 1.7e11 < 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-define 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%

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

   5. Taylor expanded in l around 0 98.0%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 98.0%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) + \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]

      2. fma-define 98.0%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)}, U\right) \]

      3. *-commutative 98.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) \cdot \ell}\right), U\right) \]

      4. *-commutative 98.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell\right), U\right) \]

      5. associate-*l* 98.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)}\right), U\right) \]

   7. Simplified 98.0%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right)\right) \cdot {\ell}^{3}\right)}, U\right) \]

   8. Taylor expanded in l around inf 98.0%

      \[\leadsto \color{blue}{0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto 0.0003968253968253968 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{7}\right)}\right) \]

      2. associate-*r* 98.0%

         \[\leadsto 0.0003968253968253968 \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot {\ell}^{7}\right)} \]

      3. associate-*l* 98.0%

         \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{7}} \]

      4. *-commutative 98.0%

         \[\leadsto \color{blue}{{\ell}^{7} \cdot \left(0.0003968253968253968 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

      5. associate-*r* 98.0%

         \[\leadsto {\ell}^{7} \cdot \color{blue}{\left(\left(0.0003968253968253968 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]

      6. associate-*r* 98.0%

         \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(0.0003968253968253968 \cdot J\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      7. *-commutative 98.0%

         \[\leadsto \left({\ell}^{7} \cdot \color{blue}{\left(J \cdot 0.0003968253968253968\right)}\right) \cdot \cos \left(0.5 \cdot K\right) \]

   10. Simplified 98.0%

       \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

   if -8.00000000000000073e106 < l < -0.0018500000000000001 or 0.00125000000000000003 < l < 1.7e11

   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 87.5%

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

   if -0.0018500000000000001 < l < 0.00125000000000000003

   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%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+106}:\\ \;\;\;\;\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -0.00185:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.00125:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 170000000000:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{t\_0}{U}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -620:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{elif}\;\ell \leq 820000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+120}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (* l (/ t_0 U))))))))
   (if (<= l -6.2e+143)
     t_1
     (if (<= l -620.0)
       (log1p (expm1 U))
       (if (<= l 820000000.0)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 3.3e+120)
           (+ U (* J (* l (+ 2.0 (* (pow K 2.0) -0.25)))))
           t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U * (1.0 + (2.0 * (J * (l * (t_0 / U)))));
	double tmp;
	if (l <= -6.2e+143) {
		tmp = t_1;
	} else if (l <= -620.0) {
		tmp = log1p(expm1(U));
	} else if (l <= 820000000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 3.3e+120) {
		tmp = U + (J * (l * (2.0 + (pow(K, 2.0) * -0.25))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U * (1.0 + (2.0 * (J * (l * (t_0 / U)))));
	double tmp;
	if (l <= -6.2e+143) {
		tmp = t_1;
	} else if (l <= -620.0) {
		tmp = Math.log1p(Math.expm1(U));
	} else if (l <= 820000000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 3.3e+120) {
		tmp = U + (J * (l * (2.0 + (Math.pow(K, 2.0) * -0.25))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U * (1.0 + (2.0 * (J * (l * (t_0 / U)))))
	tmp = 0
	if l <= -6.2e+143:
		tmp = t_1
	elif l <= -620.0:
		tmp = math.log1p(math.expm1(U))
	elif l <= 820000000.0:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 3.3e+120:
		tmp = U + (J * (l * (2.0 + (math.pow(K, 2.0) * -0.25))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l * Float64(t_0 / U))))))
	tmp = 0.0
	if (l <= -6.2e+143)
		tmp = t_1;
	elseif (l <= -620.0)
		tmp = log1p(expm1(U));
	elseif (l <= 820000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 3.3e+120)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((K ^ 2.0) * -0.25)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l * N[(t$95$0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.2e+143], t$95$1, If[LessEqual[l, -620.0], N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 820000000.0], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.3e+120], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.1999999999999998e143 or 3.29999999999999991e120 < 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-define 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%

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

   5. Taylor expanded in l around 0 100.0%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) + \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]

      2. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)}, U\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) \cdot \ell}\right), U\right) \]

      4. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell\right), U\right) \]

      5. associate-*l* 100.0%

         \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)}\right), U\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right)\right) \cdot {\ell}^{3}\right)}, U\right) \]

   8. Taylor expanded in U around inf 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto U \cdot \color{blue}{\left(\frac{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}{U} + 1\right)} \]

      2. associate-/l* 100.0%

         \[\leadsto U \cdot \left(\color{blue}{J \cdot \frac{2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)}{U}} + 1\right) \]

      3. fma-define 100.0%

         \[\leadsto U \cdot \color{blue}{\mathsf{fma}\left(J, \frac{2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)}{U}, 1\right)} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(J, \frac{\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2 \cdot \ell\right)}{U}, 1\right)} \]

   11. Taylor expanded in l around 0 49.8%

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

       1. associate-/l* 71.4%

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

       2. associate-/l* 71.4%

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

   13. Simplified 71.4%

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

   if -6.1999999999999998e143 < l < -620

   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-define 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%

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

   6. Applied egg-rr 47.7%

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

   if -620 < l < 8.2e8

   1. Initial program 68.8%

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

   3. Taylor expanded in l around 0 96.1%

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

   if 8.2e8 < l < 3.29999999999999991e120

   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 8.4%

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

      1. associate-*r* 8.4%

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

   5. Simplified 8.4%

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

   6. Taylor expanded in J around 0 8.4%

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

      1. *-commutative 8.4%

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

      2. associate-*r* 8.4%

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

      3. *-commutative 8.4%

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

      4. associate-*l* 8.4%

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

      5. *-commutative 8.4%

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

   8. Simplified 8.4%

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

   9. Taylor expanded in K around 0 46.3%

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

       1. associate-*r* 46.3%

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

       2. distribute-rgt-out 46.3%

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

       3. *-commutative 46.3%

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

   11. Simplified 46.3%

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot -0.25 + 2\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -620:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{elif}\;\ell \leq 820000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+120}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + {K}^{2} \cdot -0.25\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+125}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+195} \lor \neg \left(\ell \leq 1.75 \cdot 10^{+289}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* (pow K 2.0) -0.25)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))
   (if (<= l -1e+17)
     t_1
     (if (<= l 800000000.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 2.15e+125)
         (+ U (* J (* l t_0)))
         (if (or (<= l 3.2e+195) (not (<= l 1.75e+289)))
           t_1
           (+ U (* (* J l) t_0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 + (pow(K, 2.0) * -0.25);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -1e+17) {
		tmp = t_1;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 2.15e+125) {
		tmp = U + (J * (l * t_0));
	} else if ((l <= 3.2e+195) || !(l <= 1.75e+289)) {
		tmp = t_1;
	} else {
		tmp = U + ((J * l) * 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 = 2.0d0 + ((k ** 2.0d0) * (-0.25d0))
    t_1 = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    if (l <= (-1d+17)) then
        tmp = t_1
    else if (l <= 800000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 2.15d+125) then
        tmp = u + (j * (l * t_0))
    else if ((l <= 3.2d+195) .or. (.not. (l <= 1.75d+289))) then
        tmp = t_1
    else
        tmp = u + ((j * l) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 + (Math.pow(K, 2.0) * -0.25);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -1e+17) {
		tmp = t_1;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 2.15e+125) {
		tmp = U + (J * (l * t_0));
	} else if ((l <= 3.2e+195) || !(l <= 1.75e+289)) {
		tmp = t_1;
	} else {
		tmp = U + ((J * l) * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 + (math.pow(K, 2.0) * -0.25)
	t_1 = U * (1.0 + (2.0 * (J * (l / U))))
	tmp = 0
	if l <= -1e+17:
		tmp = t_1
	elif l <= 800000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 2.15e+125:
		tmp = U + (J * (l * t_0))
	elif (l <= 3.2e+195) or not (l <= 1.75e+289):
		tmp = t_1
	else:
		tmp = U + ((J * l) * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 + Float64((K ^ 2.0) * -0.25))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	tmp = 0.0
	if (l <= -1e+17)
		tmp = t_1;
	elseif (l <= 800000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 2.15e+125)
		tmp = Float64(U + Float64(J * Float64(l * t_0)));
	elseif ((l <= 3.2e+195) || !(l <= 1.75e+289))
		tmp = t_1;
	else
		tmp = Float64(U + Float64(Float64(J * l) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 + ((K ^ 2.0) * -0.25);
	t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	tmp = 0.0;
	if (l <= -1e+17)
		tmp = t_1;
	elseif (l <= 800000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 2.15e+125)
		tmp = U + (J * (l * t_0));
	elseif ((l <= 3.2e+195) || ~((l <= 1.75e+289)))
		tmp = t_1;
	else
		tmp = U + ((J * l) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+17], t$95$1, If[LessEqual[l, 800000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.15e+125], N[(U + N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 3.2e+195], N[Not[LessEqual[l, 1.75e+289]], $MachinePrecision]], t$95$1, N[(U + N[(N[(J * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1e17 or 2.15000000000000018e125 < l < 3.19999999999999982e195 or 1.75000000000000011e289 < 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 31.7%

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

      1. associate-*r* 31.7%

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

   5. Simplified 31.7%

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

   6. Taylor expanded in K around 0 26.2%

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

      1. associate-*r* 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in U around inf 35.8%

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

       1. associate-/l* 53.4%

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

   11. Simplified 53.4%

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

   if -1e17 < l < 8e8

   1. Initial program 69.9%

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

   3. Taylor expanded in l around 0 92.8%

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

   if 8e8 < l < 2.15000000000000018e125

   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 12.0%

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

      1. associate-*r* 12.0%

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

   5. Simplified 12.0%

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

   6. Taylor expanded in J around 0 12.0%

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

      1. *-commutative 12.0%

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

      2. associate-*r* 12.0%

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

      3. *-commutative 12.0%

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

      4. associate-*l* 12.0%

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

      5. *-commutative 12.0%

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

   8. Simplified 12.0%

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

   9. Taylor expanded in K around 0 48.9%

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

       1. associate-*r* 48.9%

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

       2. distribute-rgt-out 48.9%

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

       3. *-commutative 48.9%

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

   11. Simplified 48.9%

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

   if 3.19999999999999982e195 < l < 1.75000000000000011e289

   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 27.9%

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

      1. associate-*r* 27.9%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in K around 0 32.9%

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

      1. +-commutative 32.9%

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

      2. *-commutative 32.9%

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

      3. *-commutative 32.9%

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

      4. associate-*r* 26.7%

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

      5. associate-*l* 26.7%

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

      6. *-commutative 26.7%

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

      7. associate-*l* 26.7%

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

      8. distribute-lft-out 51.7%

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

   8. Simplified 51.7%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+17}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+125}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+195} \lor \neg \left(\ell \leq 1.75 \cdot 10^{+289}\right):\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot \left(2 + {K}^{2} \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))
   (if (<= l -5e+16)
     t_0
     (if (<= l 800000000.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 3.8e+124)
         (+ U (* J (* l (+ 2.0 (* (pow K 2.0) -0.25)))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -5e+16) {
		tmp = t_0;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 3.8e+124) {
		tmp = U + (J * (l * (2.0 + (pow(K, 2.0) * -0.25))));
	} 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 * (1.0d0 + (2.0d0 * (j * (l / u))))
    if (l <= (-5d+16)) then
        tmp = t_0
    else if (l <= 800000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 3.8d+124) then
        tmp = u + (j * (l * (2.0d0 + ((k ** 2.0d0) * (-0.25d0)))))
    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 * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -5e+16) {
		tmp = t_0;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 3.8e+124) {
		tmp = U + (J * (l * (2.0 + (Math.pow(K, 2.0) * -0.25))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (1.0 + (2.0 * (J * (l / U))))
	tmp = 0
	if l <= -5e+16:
		tmp = t_0
	elif l <= 800000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 3.8e+124:
		tmp = U + (J * (l * (2.0 + (math.pow(K, 2.0) * -0.25))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	tmp = 0.0
	if (l <= -5e+16)
		tmp = t_0;
	elseif (l <= 800000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 3.8e+124)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((K ^ 2.0) * -0.25)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (1.0 + (2.0 * (J * (l / U))));
	tmp = 0.0;
	if (l <= -5e+16)
		tmp = t_0;
	elseif (l <= 800000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 3.8e+124)
		tmp = U + (J * (l * (2.0 + ((K ^ 2.0) * -0.25))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+16], t$95$0, If[LessEqual[l, 800000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e+124], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e16 or 3.7999999999999998e124 < 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 31.1%

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

      1. associate-*r* 31.1%

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

   5. Simplified 31.1%

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

   6. Taylor expanded in K around 0 24.2%

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

      1. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in U around inf 33.2%

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

       1. associate-/l* 47.8%

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

   11. Simplified 47.8%

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

   if -5e16 < l < 8e8

   1. Initial program 69.9%

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

   3. Taylor expanded in l around 0 92.8%

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

   if 8e8 < l < 3.7999999999999998e124

   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 12.0%

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

      1. associate-*r* 12.0%

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

   5. Simplified 12.0%

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

   6. Taylor expanded in J around 0 12.0%

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

      1. *-commutative 12.0%

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

      2. associate-*r* 12.0%

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

      3. *-commutative 12.0%

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

      4. associate-*l* 12.0%

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

      5. *-commutative 12.0%

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

   8. Simplified 12.0%

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

   9. Taylor expanded in K around 0 48.9%

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

       1. associate-*r* 48.9%

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

       2. distribute-rgt-out 48.9%

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

       3. *-commutative 48.9%

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

   11. Simplified 48.9%

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot -0.25 + 2\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+124}:\\ \;\;\;\;U + \left(J \cdot 8 - J \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))
   (if (<= l -5e+16)
     t_0
     (if (<= l 800000000.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 2.75e+124) (+ U (- (* J 8.0) (* J (pow K 2.0)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -5e+16) {
		tmp = t_0;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 2.75e+124) {
		tmp = U + ((J * 8.0) - (J * pow(K, 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 * (1.0d0 + (2.0d0 * (j * (l / u))))
    if (l <= (-5d+16)) then
        tmp = t_0
    else if (l <= 800000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 2.75d+124) then
        tmp = u + ((j * 8.0d0) - (j * (k ** 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 * (1.0 + (2.0 * (J * (l / U))));
	double tmp;
	if (l <= -5e+16) {
		tmp = t_0;
	} else if (l <= 800000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 2.75e+124) {
		tmp = U + ((J * 8.0) - (J * Math.pow(K, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (1.0 + (2.0 * (J * (l / U))))
	tmp = 0
	if l <= -5e+16:
		tmp = t_0
	elif l <= 800000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 2.75e+124:
		tmp = U + ((J * 8.0) - (J * math.pow(K, 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	tmp = 0.0
	if (l <= -5e+16)
		tmp = t_0;
	elseif (l <= 800000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 2.75e+124)
		tmp = Float64(U + Float64(Float64(J * 8.0) - Float64(J * (K ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (1.0 + (2.0 * (J * (l / U))));
	tmp = 0.0;
	if (l <= -5e+16)
		tmp = t_0;
	elseif (l <= 800000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 2.75e+124)
		tmp = U + ((J * 8.0) - (J * (K ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+16], t$95$0, If[LessEqual[l, 800000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.75e+124], N[(U + N[(N[(J * 8.0), $MachinePrecision] - N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e16 or 2.74999999999999989e124 < 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 31.1%

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

      1. associate-*r* 31.1%

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

   5. Simplified 31.1%

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

   6. Taylor expanded in K around 0 24.2%

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

      1. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in U around inf 33.2%

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

       1. associate-/l* 47.8%

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

   11. Simplified 47.8%

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

   if -5e16 < l < 8e8

   1. Initial program 69.9%

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

   3. Taylor expanded in l around 0 92.8%

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

   if 8e8 < l < 2.74999999999999989e124

   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 3.4%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 36.5%

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

      1. associate-*r* 36.5%

         \[\leadsto \left(\color{blue}{\left(-1 \cdot J\right) \cdot {K}^{2}} + 8 \cdot J\right) + U \]

      2. +-commutative 36.5%

         \[\leadsto \color{blue}{\left(8 \cdot J + \left(-1 \cdot J\right) \cdot {K}^{2}\right)} + U \]

      3. associate-*r* 36.5%

         \[\leadsto \left(8 \cdot J + \color{blue}{-1 \cdot \left(J \cdot {K}^{2}\right)}\right) + U \]

      4. mul-1-neg 36.5%

         \[\leadsto \left(8 \cdot J + \color{blue}{\left(-J \cdot {K}^{2}\right)}\right) + U \]

      5. unsub-neg 36.5%

         \[\leadsto \color{blue}{\left(8 \cdot J - J \cdot {K}^{2}\right)} + U \]

      6. *-commutative 36.5%

         \[\leadsto \left(\color{blue}{J \cdot 8} - J \cdot {K}^{2}\right) + U \]

   7. Simplified 36.5%

      \[\leadsto \color{blue}{\left(J \cdot 8 - J \cdot {K}^{2}\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 800000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+124}:\\ \;\;\;\;U + \left(J \cdot 8 - J \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.36:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.36)
   (* J (+ (* 2.0 l) (/ U J)))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.36) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / 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 (cos((k / 2.0d0)) <= 0.36d0) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    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)) <= 0.36) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.36:
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.36)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.36)
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.36], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.35999999999999999

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0 61.3%

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

      1. associate-*r* 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in K around 0 35.5%

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

      1. associate-*r* 35.5%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in J around inf 42.6%

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

   if 0.35999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 83.0%

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

   3. Taylor expanded in l around 0 63.9%

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

      1. associate-*r* 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in K around 0 58.9%

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

      1. associate-*r* 58.9%

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

   8. Simplified 58.9%

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

   9. Taylor expanded in U around inf 61.3%

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

       1. associate-/l* 68.9%

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

   11. Simplified 68.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.36:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (* U (+ 1.0 (* 2.0 (* J (* l (/ (cos (* K 0.5)) U)))))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l * (cos((K * 0.5)) / 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 * (1.0d0 + (2.0d0 * (j * (l * (cos((k * 0.5d0)) / u)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l * (Math.cos((K * 0.5)) / U)))));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * (J * (l * (math.cos((K * 0.5)) / U)))))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) / U))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * (J * (l * (cos((K * 0.5)) / U)))));
end

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. associate-*l* 83.5%

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

   2. fma-define 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in l around 0 93.8%

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]
6. Step-by-step derivation

   1. distribute-lft-in 93.8%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) + \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]

   2. fma-define 93.8%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \ell \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)}, U\right) \]

   3. *-commutative 93.8%

      \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) \cdot \ell}\right), U\right) \]

   4. *-commutative 93.8%

      \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell\right), U\right) \]

   5. associate-*l* 93.8%

      \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right) + {\ell}^{2} \cdot \left(0.0003968253968253968 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) + 0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)}\right), U\right) \]

7. Simplified 93.8%

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(\ell, 2 \cdot \cos \left(0.5 \cdot K\right), \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right)\right) \cdot {\ell}^{3}\right)}, U\right) \]

8. Taylor expanded in U around inf 91.9%

   \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}{U}\right)} \]
9. Step-by-step derivation

   1. +-commutative 91.9%

      \[\leadsto U \cdot \color{blue}{\left(\frac{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}{U} + 1\right)} \]

   2. associate-/l* 92.6%

      \[\leadsto U \cdot \left(\color{blue}{J \cdot \frac{2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)}{U}} + 1\right) \]

   3. fma-define 92.6%

      \[\leadsto U \cdot \color{blue}{\mathsf{fma}\left(J, \frac{2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + {\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)}{U}, 1\right)} \]

10. Simplified 92.6%

    \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(J, \frac{\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2 \cdot \ell\right)}{U}, 1\right)} \]

11. Taylor expanded in l around 0 65.6%

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

    1. associate-/l* 71.6%

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

    2. associate-/l* 71.6%

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

13. Simplified 71.6%

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

14. Final simplification 71.6%

    \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \left(\ell \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right) \]
15. Add Preprocessing

Alternative 11: 42.9% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -6.2 \cdot 10^{+150} \lor \neg \left(J \leq 10^{+159}\right):\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -6.2e+150) (not (<= J 1e+159))) (* J (* 2.0 l)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -6.2e+150) || !(J <= 1e+159)) {
		tmp = J * (2.0 * l);
	} 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 ((j <= (-6.2d+150)) .or. (.not. (j <= 1d+159))) then
        tmp = j * (2.0d0 * l)
    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 ((J <= -6.2e+150) || !(J <= 1e+159)) {
		tmp = J * (2.0 * l);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -6.2e+150) or not (J <= 1e+159):
		tmp = J * (2.0 * l)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -6.2e+150) || !(J <= 1e+159))
		tmp = Float64(J * Float64(2.0 * l));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -6.2e+150) || ~((J <= 1e+159)))
		tmp = J * (2.0 * l);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -6.2e+150], N[Not[LessEqual[J, 1e+159]], $MachinePrecision]], N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if J < -6.20000000000000028e150 or 9.9999999999999993e158 < J

   1. Initial program 63.4%

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

   3. Taylor expanded in l around 0 85.3%

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

      1. associate-*r* 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in K around 0 57.8%

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

      1. associate-*r* 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in J around inf 50.0%

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

       1. *-commutative 50.0%

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

       2. associate-*r* 50.0%

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

       3. *-commutative 50.0%

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

   11. Simplified 50.0%

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

   if -6.20000000000000028e150 < J < 9.9999999999999993e158

   1. Initial program 89.8%

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

      1. associate-*l* 89.8%

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

      2. fma-define 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in J around 0 42.2%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.2 \cdot 10^{+150} \lor \neg \left(J \leq 10^{+159}\right):\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.4% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 95000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 95000000.0) (+ U (* l (* J 2.0))) (* J (+ (* 2.0 l) (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 95000000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((2.0 * l) + (U / J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 95000000.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = j * ((2.0d0 * l) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 95000000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((2.0 * l) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 95000000.0:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = J * ((2.0 * l) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 95000000.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 95000000.0)
		tmp = U + (l * (J * 2.0));
	else
		tmp = J * ((2.0 * l) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 95000000.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.5e7

   1. Initial program 78.0%

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

   3. Taylor expanded in l around 0 75.4%

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

      1. associate-*r* 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in K around 0 62.4%

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

      1. associate-*r* 62.4%

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

   8. Simplified 62.4%

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

   if 9.5e7 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 26.2%

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

      1. associate-*r* 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in K around 0 19.4%

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

      1. associate-*r* 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in J around inf 31.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 95000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.8% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+34} \lor \neg \left(\ell \leq 95000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.4e+34) (not (<= l 95000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.4e+34) || !(l <= 95000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-3.4d+34)) .or. (.not. (l <= 95000000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.4e+34) || !(l <= 95000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3.4e+34) or not (l <= 95000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.4e+34) || !(l <= 95000000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3.4e+34) || ~((l <= 95000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.4e+34], N[Not[LessEqual[l, 95000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -3.3999999999999999e34 or 9.5e7 < l

   1. Initial program 100.0%

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

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

   6. Applied egg-rr 15.7%

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

   if -3.3999999999999999e34 < l < 9.5e7

   1. Initial program 70.7%

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

      1. associate-*l* 70.7%

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

      2. fma-define 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in J around 0 60.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+34} \lor \neg \left(\ell \leq 95000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% 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 83.5%

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

3. Taylor expanded in l around 0 63.1%

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

   1. associate-*r* 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in K around 0 51.7%

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

   1. associate-*r* 51.7%

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

8. Simplified 51.7%

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

9. Final simplification 51.7%

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

Alternative 15: 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 83.5%

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

   1. associate-*l* 83.5%

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

   2. fma-define 83.5%

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

3. Simplified 83.5%

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

6. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

8. Simplified 2.8%

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

9. Final simplification 2.8%

   \[\leadsto 1 \]
10. Add Preprocessing

Alternative 16: 37.1% 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 83.5%

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

   1. associate-*l* 83.5%

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

   2. fma-define 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in J around 0 34.8%

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

6. Final simplification 34.8%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))