Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.6%

Time: 10.1s

Alternatives: 10

Speedup: 1.5×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -5000 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -5000.0) (not (<= t_1 0.0)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* J (* t_0 (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (J * (t_0 * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-5000.0d0)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (j * (t_0 * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (J * (t_0 * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -5000.0) or not (t_1 <= 0.0):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (J * (t_0 * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -5000.0) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -5000.0) || ~((t_1 <= 0.0)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (J * (t_0 * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000.0], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(t$95$0 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5e3 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -5e3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 69.2%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. +-commutative 99.9%

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

      11. fma-def 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 84.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)\right)} + U \]

      2. expm1-udef 74.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right)} + U \]

      3. *-commutative 74.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      4. *-commutative 74.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      5. metadata-eval 74.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      6. div-inv 74.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      7. *-commutative 74.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right)} - 1\right) + U \]

   6. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 84.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\right)} + U \]

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in l around 0 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -5000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{if}\;\ell \leq -0.017:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 14000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;U + t_0\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 (sinh l)))))
   (if (<= l -0.017)
     (+ U (* (+ (* -0.125 (* K K)) 1.0) t_0))
     (if (<= l 14000000.0)
       (+ U (* J (* (cos (/ K 2.0)) (* l 2.0))))
       (if (<= l 5.5e+102)
         (+ U t_0)
         (+
          U
          (* (cos (* K 0.5)) (* 0.3333333333333333 (* J (pow l 3.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * sinh(l));
	double tmp;
	if (l <= -0.017) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	} else if (l <= 14000000.0) {
		tmp = U + (J * (cos((K / 2.0)) * (l * 2.0)));
	} else if (l <= 5.5e+102) {
		tmp = U + t_0;
	} else {
		tmp = U + (cos((K * 0.5)) * (0.3333333333333333 * (J * pow(l, 3.0))));
	}
	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 = j * (2.0d0 * sinh(l))
    if (l <= (-0.017d0)) then
        tmp = u + ((((-0.125d0) * (k * k)) + 1.0d0) * t_0)
    else if (l <= 14000000.0d0) then
        tmp = u + (j * (cos((k / 2.0d0)) * (l * 2.0d0)))
    else if (l <= 5.5d+102) then
        tmp = u + t_0
    else
        tmp = u + (cos((k * 0.5d0)) * (0.3333333333333333d0 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * Math.sinh(l));
	double tmp;
	if (l <= -0.017) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	} else if (l <= 14000000.0) {
		tmp = U + (J * (Math.cos((K / 2.0)) * (l * 2.0)));
	} else if (l <= 5.5e+102) {
		tmp = U + t_0;
	} else {
		tmp = U + (Math.cos((K * 0.5)) * (0.3333333333333333 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * math.sinh(l))
	tmp = 0
	if l <= -0.017:
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0)
	elif l <= 14000000.0:
		tmp = U + (J * (math.cos((K / 2.0)) * (l * 2.0)))
	elif l <= 5.5e+102:
		tmp = U + t_0
	else:
		tmp = U + (math.cos((K * 0.5)) * (0.3333333333333333 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * sinh(l)))
	tmp = 0.0
	if (l <= -0.017)
		tmp = Float64(U + Float64(Float64(Float64(-0.125 * Float64(K * K)) + 1.0) * t_0));
	elseif (l <= 14000000.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K / 2.0)) * Float64(l * 2.0))));
	elseif (l <= 5.5e+102)
		tmp = Float64(U + t_0);
	else
		tmp = Float64(U + Float64(cos(Float64(K * 0.5)) * Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * sinh(l));
	tmp = 0.0;
	if (l <= -0.017)
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	elseif (l <= 14000000.0)
		tmp = U + (J * (cos((K / 2.0)) * (l * 2.0)));
	elseif (l <= 5.5e+102)
		tmp = U + t_0;
	else
		tmp = U + (cos((K * 0.5)) * (0.3333333333333333 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -0.017], N[(U + N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 14000000.0], N[(U + N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e+102], N[(U + t$95$0), $MachinePrecision], N[(U + N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -0.017000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 1.7%

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

      1. associate-*r* 1.7%

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

      2. distribute-rgt1-in 83.9%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 83.9%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 83.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 26.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 26.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 26.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   6. Applied egg-rr 38.6%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 26.0%

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

      2. expm1-log1p 64.9%

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

   8. Simplified 83.9%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} + U \]

   if -0.017000000000000001 < l < 1.4e7

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l* 99.1%

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

      7. *-commutative 99.1%

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

      8. associate-*r* 99.1%

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

      9. distribute-rgt-out 99.1%

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

      10. +-commutative 99.1%

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

      11. fma-def 99.1%

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

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)\right)} + U \]

      2. expm1-udef 74.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right)} + U \]

      3. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      4. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      5. metadata-eval 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      6. div-inv 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      7. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right)} - 1\right) + U \]

   6. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\right)} + U \]

      2. expm1-log1p 99.1%

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

      3. associate-*l* 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in l around 0 99.2%

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

       1. *-commutative 99.2%

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

   11. Simplified 99.2%

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

   if 1.4e7 < l < 5.49999999999999981e102

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 95.0%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 45.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 45.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 45.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   4. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 45.0%

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

      2. expm1-log1p 95.0%

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

   6. Simplified 95.0%

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

   if 5.49999999999999981e102 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.017:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(J \cdot \left(2 \cdot \sinh \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 14000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 86.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.25) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.25) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.25)
		tmp = U + (J * (t_0 * (l * 2.0)));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.25

   1. Initial program 87.4%

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

   2. Taylor expanded in l around 0 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-*l* 90.5%

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

      3. *-commutative 90.5%

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

      4. associate-*r* 90.5%

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

      5. *-commutative 90.5%

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

      6. associate-*l* 90.5%

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

      7. *-commutative 90.5%

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

      8. associate-*r* 90.5%

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

      9. distribute-rgt-out 90.5%

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

      10. +-commutative 90.5%

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

      11. fma-def 90.5%

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

   4. Simplified 90.5%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 64.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)\right)} + U \]

      2. expm1-udef 58.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right)} + U \]

      3. *-commutative 58.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      4. *-commutative 58.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      5. metadata-eval 58.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      6. div-inv 58.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      7. *-commutative 58.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right)} - 1\right) + U \]

   6. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\right)} + U \]

      2. expm1-log1p 90.5%

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

      3. associate-*l* 90.5%

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

   8. Simplified 90.5%

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

   9. Taylor expanded in l around 0 67.7%

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

       1. *-commutative 67.7%

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

   11. Simplified 67.7%

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

   if 0.25 < (cos.f64 (/.f64 K 2))

   1. Initial program 83.9%

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

   2. Taylor expanded in K around 0 83.5%

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

      1. expm1-log1p-u 55.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 55.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 55.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 55.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   4. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 59.4%

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

      2. expm1-log1p 94.5%

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

   6. Simplified 94.5%

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

4. Final simplification 87.1%

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

Alternative 4: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.55)
   (+ U (* (+ (* -0.125 (* K K)) 1.0) (* J (* l 2.0))))
   (+ U (* J (* 2.0 (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.55) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.55d0)) then
        tmp = u + ((((-0.125d0) * (k * k)) + 1.0d0) * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    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)) <= -0.55) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.55:
		tmp = U + (((-0.125 * (K * K)) + 1.0) * (J * (l * 2.0)))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.55)
		tmp = Float64(U + Float64(Float64(Float64(-0.125 * Float64(K * K)) + 1.0) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.55)
		tmp = U + (((-0.125 * (K * K)) + 1.0) * (J * (l * 2.0)));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.55], N[(U + N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.55000000000000004

   1. Initial program 86.9%

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

   2. Taylor expanded in K around 0 20.1%

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

      1. associate-*r* 20.1%

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

      2. distribute-rgt1-in 81.6%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 81.6%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 81.6%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

   5. Taylor expanded in l around 0 59.4%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J\right) + U \]

   if -0.55000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.5%

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

   2. Taylor expanded in K around 0 79.1%

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

      1. expm1-log1p-u 54.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 54.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 54.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 54.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   4. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 58.1%

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

      2. expm1-log1p 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 5: 86.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 14000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 (sinh l)))))
   (if (<= l -4.0)
     (+ U (* (+ (* -0.125 (* K K)) 1.0) t_0))
     (if (<= l 14000000.0)
       (+ U (* J (* (cos (/ K 2.0)) (* l 2.0))))
       (+ U t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * sinh(l));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	} else if (l <= 14000000.0) {
		tmp = U + (J * (cos((K / 2.0)) * (l * 2.0)));
	} else {
		tmp = U + 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 = j * (2.0d0 * sinh(l))
    if (l <= (-4.0d0)) then
        tmp = u + ((((-0.125d0) * (k * k)) + 1.0d0) * t_0)
    else if (l <= 14000000.0d0) then
        tmp = u + (j * (cos((k / 2.0d0)) * (l * 2.0d0)))
    else
        tmp = u + t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * Math.sinh(l));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	} else if (l <= 14000000.0) {
		tmp = U + (J * (Math.cos((K / 2.0)) * (l * 2.0)));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * math.sinh(l))
	tmp = 0
	if l <= -4.0:
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0)
	elif l <= 14000000.0:
		tmp = U + (J * (math.cos((K / 2.0)) * (l * 2.0)))
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * sinh(l)))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(U + Float64(Float64(Float64(-0.125 * Float64(K * K)) + 1.0) * t_0));
	elseif (l <= 14000000.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K / 2.0)) * Float64(l * 2.0))));
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * sinh(l));
	tmp = 0.0;
	if (l <= -4.0)
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	elseif (l <= 14000000.0)
		tmp = U + (J * (cos((K / 2.0)) * (l * 2.0)));
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.0], N[(U + N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 14000000.0], N[(U + N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 1.7%

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

      1. associate-*r* 1.7%

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

      2. distribute-rgt1-in 83.9%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 83.9%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 83.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 26.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 26.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 26.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   6. Applied egg-rr 38.6%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 26.0%

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

      2. expm1-log1p 64.9%

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

   8. Simplified 83.9%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} + U \]

   if -4 < l < 1.4e7

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l* 99.1%

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

      7. *-commutative 99.1%

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

      8. associate-*r* 99.1%

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

      9. distribute-rgt-out 99.1%

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

      10. +-commutative 99.1%

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

      11. fma-def 99.1%

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

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} + U \]
   5. Step-by-step derivation

      1. expm1-log1p-u 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)\right)} + U \]

      2. expm1-udef 74.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right)} + U \]

      3. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      4. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      5. metadata-eval 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      6. div-inv 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} - 1\right) + U \]

      7. *-commutative 74.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right)} - 1\right) + U \]

   6. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)} - 1\right)} + U \]
   7. Step-by-step derivation

      1. expm1-def 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\right)} + U \]

      2. expm1-log1p 99.1%

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

      3. associate-*l* 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in l around 0 99.2%

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

       1. *-commutative 99.2%

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

   11. Simplified 99.2%

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

   if 1.4e7 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 80.3%

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

      1. expm1-log1p-u 36.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 36.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. *-commutative 36.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}\right)} - 1\right) + U \]

      4. sinh-undef 36.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right)} - 1\right) + U \]

   4. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 36.4%

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

      2. expm1-log1p 80.3%

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

   6. Simplified 80.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(J \cdot \left(2 \cdot \sinh \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 14000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 6: 58.7% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-27}:\\ \;\;\;\;U + t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l 2.0))))
   (if (<= l -1.9e-38)
     (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
     (if (<= l 5e-27) (+ U t_0) (+ U (* (+ (* -0.125 (* K K)) 1.0) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= -1.9e-38) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (l <= 5e-27) {
		tmp = U + t_0;
	} else {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * 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 = j * (l * 2.0d0)
    if (l <= (-1.9d-38)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else if (l <= 5d-27) then
        tmp = u + t_0
    else
        tmp = u + ((((-0.125d0) * (k * k)) + 1.0d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= -1.9e-38) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (l <= 5e-27) {
		tmp = U + t_0;
	} else {
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * 2.0)
	tmp = 0
	if l <= -1.9e-38:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	elif l <= 5e-27:
		tmp = U + t_0
	else:
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (l <= -1.9e-38)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	elseif (l <= 5e-27)
		tmp = Float64(U + t_0);
	else
		tmp = Float64(U + Float64(Float64(Float64(-0.125 * Float64(K * K)) + 1.0) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * 2.0);
	tmp = 0.0;
	if (l <= -1.9e-38)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	elseif (l <= 5e-27)
		tmp = U + t_0;
	else
		tmp = U + (((-0.125 * (K * K)) + 1.0) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e-38], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e-27], N[(U + t$95$0), $MachinePrecision], N[(U + N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9e-38

   1. Initial program 98.4%

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

   2. Taylor expanded in l around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-*l* 74.0%

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

      3. *-commutative 74.0%

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

      4. associate-*r* 74.0%

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

      5. *-commutative 74.0%

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

      6. associate-*l* 74.0%

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

      7. *-commutative 74.0%

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

      8. associate-*r* 74.0%

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

      9. distribute-rgt-out 74.0%

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

      10. +-commutative 74.0%

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

      11. fma-def 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in l around 0 31.2%

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

      1. *-commutative 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in K around 0 10.1%

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

      1. +-commutative 10.1%

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

      2. unpow2 10.1%

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

      3. associate-*r* 10.1%

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

      4. distribute-rgt-out 33.9%

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

   10. Simplified 33.9%

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

   if -1.9e-38 < l < 5.0000000000000002e-27

   1. Initial program 69.5%

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

   2. Taylor expanded in K around 0 69.5%

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

   3. Taylor expanded in l around 0 86.2%

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

   if 5.0000000000000002e-27 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 2.9%

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

      1. associate-*r* 2.9%

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

      2. distribute-rgt1-in 72.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 72.5%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 72.5%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

   5. Taylor expanded in l around 0 32.3%

      \[\leadsto \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-27}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 58.7% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38} \lor \neg \left(\ell \leq 5 \cdot 10^{-27}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.9e-38) (not (<= l 5e-27)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e-38) || !(l <= 5e-27)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.9d-38)) .or. (.not. (l <= 5d-27))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e-38) || !(l <= 5e-27)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.9e-38) or not (l <= 5e-27):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.9e-38) || !(l <= 5e-27))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.9e-38) || ~((l <= 5e-27)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.9e-38], N[Not[LessEqual[l, 5e-27]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9e-38 or 5.0000000000000002e-27 < l

   1. Initial program 99.3%

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

   2. Taylor expanded in l around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*l* 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r* 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-*l* 75.1%

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

      7. *-commutative 75.1%

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

      8. associate-*r* 75.1%

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

      9. distribute-rgt-out 75.1%

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

      10. +-commutative 75.1%

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

      11. fma-def 75.1%

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

   4. Simplified 75.1%

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

   5. Taylor expanded in l around 0 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in K around 0 10.4%

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

      1. +-commutative 10.4%

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

      2. unpow2 10.4%

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

      3. associate-*r* 10.4%

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

      4. distribute-rgt-out 32.4%

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

   10. Simplified 32.4%

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

   if -1.9e-38 < l < 5.0000000000000002e-27

   1. Initial program 69.5%

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

   2. Taylor expanded in K around 0 69.5%

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

   3. Taylor expanded in l around 0 86.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38} \lor \neg \left(\ell \leq 5 \cdot 10^{-27}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 43.5% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38} \lor \neg \left(\ell \leq 38000000\right):\\ \;\;\;\;U + J \cdot \left(K \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.9e-38) (not (<= l 38000000.0))) (+ U (* J (* K K))) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e-38) || !(l <= 38000000.0)) {
		tmp = U + (J * (K * K));
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.9d-38)) .or. (.not. (l <= 38000000.0d0))) then
        tmp = u + (j * (k * k))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e-38) || !(l <= 38000000.0)) {
		tmp = U + (J * (K * K));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.9e-38) or not (l <= 38000000.0):
		tmp = U + (J * (K * K))
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.9e-38) || !(l <= 38000000.0))
		tmp = Float64(U + Float64(J * Float64(K * K)));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.9e-38) || ~((l <= 38000000.0)))
		tmp = U + (J * (K * K));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.9e-38], N[Not[LessEqual[l, 38000000.0]], $MachinePrecision]], N[(U + N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.9e-38 or 3.8e7 < l

   1. Initial program 99.2%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in K around 0 17.2%

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

      1. distribute-rgt-out 17.2%

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

      2. unpow2 17.2%

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

   6. Simplified 17.2%

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

   7. Taylor expanded in K around inf 17.0%

      \[\leadsto \color{blue}{{K}^{2} \cdot J} + U \]
   8. Step-by-step derivation

      1. unpow2 17.0%

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

      2. *-commutative 17.0%

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

   9. Simplified 17.0%

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

   if -1.9e-38 < l < 3.8e7

   1. Initial program 70.3%

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

   2. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l* 99.1%

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

      7. *-commutative 99.1%

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

      8. associate-*r* 99.1%

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

      9. distribute-rgt-out 99.1%

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

      10. +-commutative 99.1%

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

      11. fma-def 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in J around 0 69.5%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-38} \lor \neg \left(\ell \leq 38000000\right):\\ \;\;\;\;U + J \cdot \left(K \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9: 53.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in K around 0 70.9%

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

3. Taylor expanded in l around 0 51.9%

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

4. Final simplification 51.9%

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

Alternative 10: 36.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in l around 0 87.1%

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

   1. *-commutative 87.1%

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

   2. associate-*l* 87.1%

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

   3. *-commutative 87.1%

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

   4. associate-*r* 87.1%

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

   5. *-commutative 87.1%

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

   6. associate-*l* 87.1%

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

   7. *-commutative 87.1%

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

   8. associate-*r* 87.1%

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

   9. distribute-rgt-out 87.1%

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

   10. +-commutative 87.1%

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

   11. fma-def 87.1%

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

4. Simplified 87.1%

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

5. Taylor expanded in J around 0 35.6%

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

6. Final simplification 35.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023194 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))