Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.9%

Time: 15.0s

Alternatives: 22

Speedup: 2.1×

• 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: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sinh-undef N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sinh-lowering-sinh.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. div-inv N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 45.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot J\right)\\ t_1 := \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-158}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* l J)))
        (t_1 (* (cos (/ K 2.0)) (* J (- (exp l) (exp (- l)))))))
   (if (<= t_1 -1e-59) t_0 (if (<= t_1 4e-158) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double t_1 = cos((K / 2.0)) * (J * (exp(l) - exp(-l)));
	double tmp;
	if (t_1 <= -1e-59) {
		tmp = t_0;
	} else if (t_1 <= 4e-158) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 * (l * j)
    t_1 = cos((k / 2.0d0)) * (j * (exp(l) - exp(-l)))
    if (t_1 <= (-1d-59)) then
        tmp = t_0
    else if (t_1 <= 4d-158) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double t_1 = Math.cos((K / 2.0)) * (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (t_1 <= -1e-59) {
		tmp = t_0;
	} else if (t_1 <= 4e-158) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (l * J)
	t_1 = math.cos((K / 2.0)) * (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if t_1 <= -1e-59:
		tmp = t_0
	elif t_1 <= 4e-158:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(l * J))
	t_1 = Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (t_1 <= -1e-59)
		tmp = t_0;
	elseif (t_1 <= 4e-158)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (l * J);
	t_1 = cos((K / 2.0)) * (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (t_1 <= -1e-59)
		tmp = t_0;
	elseif (t_1 <= 4e-158)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-59], t$95$0, If[LessEqual[t$95$1, 4e-158], U, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -1e-59 or 4.00000000000000026e-158 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.2%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 71.9

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

   5. Simplified 71.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      13. *-lowering-*.f64 53.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. *-lowering-*.f64 26.5

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

   11. Simplified 26.5%

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

   12. Taylor expanded in J around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-lowering-*.f64 26.5

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

   14. Simplified 26.5%

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

   if -1e-59 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 4.00000000000000026e-158

   1. Initial program 70.4%

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

   3. Taylor expanded in J around 0

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

   5. Simplified 69.8%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \leq -1 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \leq 4 \cdot 10^{-158}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-59}:\\ \;\;\;\;J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 -1e-59)
     (* J (* l (fma l (* l 0.3333333333333333) 2.0)))
     (if (<= t_0 1e+186)
       (fma 2.0 (* l J) U)
       (* J (* 0.3333333333333333 (* l (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -1e-59) {
		tmp = J * (l * fma(l, (l * 0.3333333333333333), 2.0));
	} else if (t_0 <= 1e+186) {
		tmp = fma(2.0, (l * J), U);
	} else {
		tmp = J * (0.3333333333333333 * (l * (l * l)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= -1e-59)
		tmp = Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)));
	elseif (t_0 <= 1e+186)
		tmp = fma(2.0, Float64(l * J), U);
	else
		tmp = Float64(J * Float64(0.3333333333333333 * Float64(l * Float64(l * l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-59], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+186], N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1e-59

   1. Initial program 99.4%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in J around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2\right)\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 66.4

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

   8. Simplified 66.4%

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

   9. Taylor expanded in K around 0

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right)\right) \]

       4. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right) \]

       6. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2\right)\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right)\right) \]

       9. *-lowering-*.f64 57.4

          \[\leadsto J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right)\right) \]

   11. Simplified 57.4%

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)} \]

   if -1e-59 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 9.9999999999999998e185

   1. Initial program 70.4%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      13. *-lowering-*.f64 84.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. *-lowering-*.f64 84.0

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

   11. Simplified 84.0%

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

   if 9.9999999999999998e185 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 75.7

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

   5. Simplified 75.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      13. *-lowering-*.f64 50.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 50.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \frac{1}{3}\right)} \cdot {\ell}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{{\ell}^{2}} \cdot \ell\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \]

       8. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right) \]

       10. unpow3 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{{\ell}^{3}}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       12. cube-mult N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right) \]

       15. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

       16. *-lowering-*.f64 57.3

           \[\leadsto J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   11. Simplified 57.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -1 \cdot 10^{-59}:\\ \;\;\;\;J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\\ \mathbf{elif}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.12:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.12)
     (+ U (* t_0 (* J (* l (fma l (* l 0.3333333333333333) 2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.12) {
		tmp = U + (t_0 * (J * (l * fma(l, (l * 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.12)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return 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, 0.12], N[(U + N[(t$95$0 * N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.12

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 87.1

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

   5. Simplified 87.1%

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

   if 0.12 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 95.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.12:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.12)
   (fma l (* (fma l (* l 0.3333333333333333) 2.0) (* J (cos (* K 0.5)))) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.12) {
		tmp = fma(l, (fma(l, (l * 0.3333333333333333), 2.0) * (J * cos((K * 0.5)))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.12)
		tmp = fma(l, Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * Float64(J * cos(Float64(K * 0.5)))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.12], N[(l * N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.12

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 84.6%

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

   if 0.12 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 95.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(2, \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma 2.0 (* (cos (* K 0.5)) (* l J)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(2.0, (cos((K * 0.5)) * (l * J)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(2.0, Float64(cos(Float64(K * 0.5)) * Float64(l * J)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(2.0 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 81.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 76.5

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

   7. Simplified 76.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(2, \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), K \cdot \left(K \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       l
       (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
       0.3333333333333333)
      2.0))
    (* K (* K (fma J -0.125 (/ J (* K K)))))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), (K * (K * fma(J, -0.125, (J / (K * K))))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), Float64(K * Float64(K * fma(J, -0.125, Float64(J / Float64(K * K))))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * N[(J * -0.125 + N[(J / N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. +-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 95.6

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

   5. Simplified 95.6%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2} + J\right)} + U \]

      5. +-commutative N/A

         \[\leadsto \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \color{blue}{\left(J + \left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right)} + U \]

   8. Simplified 47.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \mathsf{fma}\left(J, -0.125 \cdot \left(K \cdot K\right), J\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right)}, U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot {K}^{2}}, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot \color{blue}{\left(K \cdot K\right)}, U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K}, U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K}, U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right)} \cdot K, U\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\left(\color{blue}{J \cdot \frac{-1}{8}} + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K, U\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\color{blue}{\mathsf{fma}\left(J, \frac{-1}{8}, \frac{J}{{K}^{2}}\right)} \cdot K\right) \cdot K, U\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\mathsf{fma}\left(J, \frac{-1}{8}, \color{blue}{\frac{J}{{K}^{2}}}\right) \cdot K\right) \cdot K, U\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\mathsf{fma}\left(J, \frac{-1}{8}, \frac{J}{\color{blue}{K \cdot K}}\right) \cdot K\right) \cdot K, U\right) \]

       10. *-lowering-*.f64 52.6

           \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \left(\mathsf{fma}\left(J, -0.125, \frac{J}{\color{blue}{K \cdot K}}\right) \cdot K\right) \cdot K, U\right) \]

   11. Simplified 52.6%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \color{blue}{\left(\mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right) \cdot K\right) \cdot K}, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), K \cdot \left(K \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. sinh-undef N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. div-inv N/A

      \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot 2, \sinh \ell \cdot J, U\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot 2, \sinh \ell \cdot J, U\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot \color{blue}{\frac{1}{2}}\right) \cdot 2, \sinh \ell \cdot J, U\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot 2, \color{blue}{J \cdot \sinh \ell}, U\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot 2, \color{blue}{J \cdot \sinh \ell}, U\right) \]

   14. sinh-lowering-sinh.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(2 \cdot \cos \left(K \cdot 0.5\right), \sinh \ell \cdot J, U\right) \]
6. Add Preprocessing

Alternative 9: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot t\_0, K \cdot \left(K \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \ell \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (fma
          (* l l)
          (fma
           l
           (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
           0.3333333333333333)
          2.0)))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma (* l t_0) (* K (* K (fma J -0.125 (/ J (* K K))))) U)
     (fma t_0 (* l J) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0);
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * t_0), (K * (K * fma(J, -0.125, (J / (K * K))))), U);
	} else {
		tmp = fma(t_0, (l * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * t_0), Float64(K * Float64(K * fma(J, -0.125, Float64(J / Float64(K * K))))), U);
	else
		tmp = fma(t_0, Float64(l * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(l * t$95$0), $MachinePrecision] * N[(K * N[(K * N[(J * -0.125 + N[(J / N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(t$95$0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. +-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 95.6

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

   5. Simplified 95.6%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2} + J\right)} + U \]

      5. +-commutative N/A

         \[\leadsto \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \color{blue}{\left(J + \left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right)} + U \]

   8. Simplified 47.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \mathsf{fma}\left(J, -0.125 \cdot \left(K \cdot K\right), J\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right)}, U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot {K}^{2}}, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot \color{blue}{\left(K \cdot K\right)}, U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K}, U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K}, U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot J + \frac{J}{{K}^{2}}\right) \cdot K\right)} \cdot K, U\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\left(\color{blue}{J \cdot \frac{-1}{8}} + \frac{J}{{K}^{2}}\right) \cdot K\right) \cdot K, U\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\color{blue}{\mathsf{fma}\left(J, \frac{-1}{8}, \frac{J}{{K}^{2}}\right)} \cdot K\right) \cdot K, U\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\mathsf{fma}\left(J, \frac{-1}{8}, \color{blue}{\frac{J}{{K}^{2}}}\right) \cdot K\right) \cdot K, U\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right), \left(\mathsf{fma}\left(J, \frac{-1}{8}, \frac{J}{\color{blue}{K \cdot K}}\right) \cdot K\right) \cdot K, U\right) \]

       10. *-lowering-*.f64 52.6

           \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \left(\mathsf{fma}\left(J, -0.125, \frac{J}{\color{blue}{K \cdot K}}\right) \cdot K\right) \cdot K, U\right) \]

   11. Simplified 52.6%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \color{blue}{\left(\mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right) \cdot K\right) \cdot K}, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. +-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right), J \cdot \ell, U\right)} \]

   8. Simplified 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J \cdot \ell, U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), K \cdot \left(K \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \ell \cdot J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-69)
   (fma (* 2.0 (sinh l)) J U)
   (+
    U
    (*
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma
         l
         (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
         0.3333333333333333)
        2.0)))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-69) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = U + ((J * (l * fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos((K / 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-69)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-69], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 9.9999999999999996e-70

   1. Initial program 81.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 87.4%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   if 9.9999999999999996e-70 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. +-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.4

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

   5. Simplified 96.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-69)
   (fma (* 2.0 (sinh l)) J U)
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma (* l l) 0.016666666666666666 0.3333333333333333)
        2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-69) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-69)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-69], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 9.9999999999999996e-70

   1. Initial program 81.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 87.4%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   if 9.9999999999999996e-70 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{1}{60} \cdot {\ell}^{2} + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \frac{1}{60}} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{60}, \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{60}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 95.2

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

   5. Simplified 95.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := J \cdot \left(t\_0 \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ t_2 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.003:\\ \;\;\;\;U + \frac{J}{\frac{1}{t\_2}}\\ \mathbf{elif}\;\ell \leq 0.00043:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* J (* t_0 (* 0.3333333333333333 (* l (* l l))))))
        (t_2 (* 2.0 (sinh l))))
   (if (<= l -1e+113)
     t_1
     (if (<= l -0.003)
       (+ U (/ J (/ 1.0 t_2)))
       (if (<= l 0.00043)
         (fma (* t_0 (* 2.0 l)) J U)
         (if (<= l 5.5e+102) (fma t_2 J U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = J * (t_0 * (0.3333333333333333 * (l * (l * l))));
	double t_2 = 2.0 * sinh(l);
	double tmp;
	if (l <= -1e+113) {
		tmp = t_1;
	} else if (l <= -0.003) {
		tmp = U + (J / (1.0 / t_2));
	} else if (l <= 0.00043) {
		tmp = fma((t_0 * (2.0 * l)), J, U);
	} else if (l <= 5.5e+102) {
		tmp = fma(t_2, J, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(J * Float64(t_0 * Float64(0.3333333333333333 * Float64(l * Float64(l * l)))))
	t_2 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (l <= -1e+113)
		tmp = t_1;
	elseif (l <= -0.003)
		tmp = Float64(U + Float64(J / Float64(1.0 / t_2)));
	elseif (l <= 0.00043)
		tmp = fma(Float64(t_0 * Float64(2.0 * l)), J, U);
	elseif (l <= 5.5e+102)
		tmp = fma(t_2, J, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(t$95$0 * N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+113], t$95$1, If[LessEqual[l, -0.003], N[(U + N[(J / N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00043], N[(N[(t$95$0 * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 5.5e+102], N[(t$95$2 * J + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1e113 or 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. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in J around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2\right)\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in l around inf

      \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]

       2. unpow3 N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       3. unpow2 N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot \left(\color{blue}{{\ell}^{2}} \cdot \ell\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       5. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right)} \]

       7. cos-lowering-cos.f64 N/A

          \[\leadsto J \cdot \left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right)\right) \]

       11. unpow3 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{{\ell}^{3}}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)}\right) \]

       13. cube-mult N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

   if -1e113 < l < -0.0030000000000000001

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. div-inv N/A

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. clear-num N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \frac{\color{blue}{J}}{\frac{1}{2 \cdot \sinh \ell}} + U \]
   6. Step-by-step derivation

   7. Simplified 80.8%

      \[\leadsto \frac{\color{blue}{J}}{\frac{1}{2 \cdot \sinh \ell}} + U \]

   if -0.0030000000000000001 < l < 4.29999999999999989e-4

   1. Initial program 70.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 4.29999999999999989e-4 < l < 5.49999999999999981e102

   1. Initial program 99.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 72.2%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+113}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.003:\\ \;\;\;\;U + \frac{J}{\frac{1}{2 \cdot \sinh \ell}}\\ \mathbf{elif}\;\ell \leq 0.00043:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := J \cdot \left(t\_1 \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -0.00037:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* 2.0 (sinh l)) J U))
        (t_1 (cos (* K 0.5)))
        (t_2 (* J (* t_1 (* 0.3333333333333333 (* l (* l l)))))))
   (if (<= l -4.8e+112)
     t_2
     (if (<= l -0.00037)
       t_0
       (if (<= l 8.8e-5)
         (fma (* t_1 (* 2.0 l)) J U)
         (if (<= l 5.5e+102) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((2.0 * sinh(l)), J, U);
	double t_1 = cos((K * 0.5));
	double t_2 = J * (t_1 * (0.3333333333333333 * (l * (l * l))));
	double tmp;
	if (l <= -4.8e+112) {
		tmp = t_2;
	} else if (l <= -0.00037) {
		tmp = t_0;
	} else if (l <= 8.8e-5) {
		tmp = fma((t_1 * (2.0 * l)), J, U);
	} else if (l <= 5.5e+102) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(2.0 * sinh(l)), J, U)
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(J * Float64(t_1 * Float64(0.3333333333333333 * Float64(l * Float64(l * l)))))
	tmp = 0.0
	if (l <= -4.8e+112)
		tmp = t_2;
	elseif (l <= -0.00037)
		tmp = t_0;
	elseif (l <= 8.8e-5)
		tmp = fma(Float64(t_1 * Float64(2.0 * l)), J, U);
	elseif (l <= 5.5e+102)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(J * N[(t$95$1 * N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+112], t$95$2, If[LessEqual[l, -0.00037], t$95$0, If[LessEqual[l, 8.8e-5], N[(N[(t$95$1 * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 5.5e+102], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8e112 or 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. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in J around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2\right)\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in l around inf

      \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]

       2. unpow3 N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       3. unpow2 N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot \left(\color{blue}{{\ell}^{2}} \cdot \ell\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \]

       5. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right)} \]

       7. cos-lowering-cos.f64 N/A

          \[\leadsto J \cdot \left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right)\right) \]

       11. unpow3 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{{\ell}^{3}}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)}\right) \]

       13. cube-mult N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

   if -4.8e112 < l < -3.6999999999999999e-4 or 8.7999999999999998e-5 < l < 5.49999999999999981e102

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 76.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   if -3.6999999999999999e-4 < l < 8.7999999999999998e-5

   1. Initial program 70.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00037:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.85 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 1.85e+148)
   (fma (* 2.0 (sinh l)) J U)
   (* J (* (cos (* K 0.5)) (* 2.0 l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 1.85e+148) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = J * (cos((K * 0.5)) * (2.0 * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 1.85e+148)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(2.0 * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 1.85e+148], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.8500000000000001e148

   1. Initial program 87.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sinh-undef N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sinh-lowering-sinh.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 84.3%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   if 1.8500000000000001e148 < J

   1. Initial program 58.9%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in J around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2\right)\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 94.1

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

   8. Simplified 94.1%

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

   9. Taylor expanded in l around 0

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

   11. Simplified 91.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.85 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \ell \cdot J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  (fma
   (* l l)
   (fma
    l
    (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
    0.3333333333333333)
   2.0)
  (* l J)
  U))

C

double code(double J, double l, double K, double U) {
	return fma(fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0), (l * J), U);
}

Julia

function code(J, l, K, U)
	return fma(fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0), Float64(l * J), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * J), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. +-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. *-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. *-lowering-*.f64 96.2

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

5. Simplified 96.2%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right), J \cdot \ell, U\right)} \]

8. Simplified 79.3%

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

9. Final simplification 79.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \ell \cdot J, U\right) \]
10. Add Preprocessing

Alternative 16: 75.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  l
  (*
   J
   (fma
    (* l l)
    (fma
     (* l l)
     (fma (* l l) 0.0003968253968253968 0.016666666666666666)
     0.3333333333333333)
    2.0))
  U))

C

double code(double J, double l, double K, double U) {
	return fma(l, (J * fma((l * l), fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)), U);
}

Julia

function code(J, l, K, U)
	return fma(l, Float64(J * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(l * N[(J * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. +-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. *-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. *-lowering-*.f64 96.2

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

5. Simplified 96.2%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   4. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2} + J\right)} + U \]

   5. +-commutative N/A

      \[\leadsto \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \color{blue}{\left(J + \left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}\right)} + U \]

8. Simplified 59.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), \mathsf{fma}\left(J, -0.125 \cdot \left(K \cdot K\right), J\right), U\right)} \]

9. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot J} + U \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{\ell \cdot \left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot J\right)} + U \]

    4. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot J, U\right)} \]

11. Simplified 79.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)} \]
12. Add Preprocessing

Alternative 17: 75.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(0.0003968253968253968 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right), 2\right), \ell \cdot J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  (fma (* l l) (* l (* 0.0003968253968253968 (* l (* l l)))) 2.0)
  (* l J)
  U))

C

double code(double J, double l, double K, double U) {
	return fma(fma((l * l), (l * (0.0003968253968253968 * (l * (l * l)))), 2.0), (l * J), U);
}

Julia

function code(J, l, K, U)
	return fma(fma(Float64(l * l), Float64(l * Float64(0.0003968253968253968 * Float64(l * Float64(l * l)))), 2.0), Float64(l * J), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(0.0003968253968253968 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * J), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. +-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. *-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. *-lowering-*.f64 96.2

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

5. Simplified 96.2%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right), J \cdot \ell, U\right)} \]

8. Simplified 79.3%

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

9. Taylor expanded in l around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{1}{2520} \cdot {\ell}^{4}}, 2\right), J \cdot \ell, U\right) \]
10. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{2520} \cdot {\ell}^{\color{blue}{\left(2 \cdot 2\right)}}, 2\right), J \cdot \ell, U\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{2520} \cdot \color{blue}{\left({\ell}^{2} \cdot {\ell}^{2}\right)}, 2\right), J \cdot \ell, U\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}}, 2\right), J \cdot \ell, U\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{2520} \cdot {\ell}^{2}\right)}, 2\right), J \cdot \ell, U\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{2520} \cdot {\ell}^{2}\right), 2\right), J \cdot \ell, U\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)}, 2\right), J \cdot \ell, U\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \color{blue}{\left(\left(\frac{1}{2520} \cdot {\ell}^{2}\right) \cdot \ell\right)}, 2\right), J \cdot \ell, U\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot \left({\ell}^{2} \cdot \ell\right)\right)}, 2\right), J \cdot \ell, U\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\frac{1}{2520} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right), 2\right), J \cdot \ell, U\right) \]

    10. unpow3 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\frac{1}{2520} \cdot \color{blue}{{\ell}^{3}}\right), 2\right), J \cdot \ell, U\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{3}\right)}, 2\right), J \cdot \ell, U\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \color{blue}{\left({\ell}^{3} \cdot \frac{1}{2520}\right)}, 2\right), J \cdot \ell, U\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \color{blue}{\left({\ell}^{3} \cdot \frac{1}{2520}\right)}, 2\right), J \cdot \ell, U\right) \]

    14. cube-mult N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)} \cdot \frac{1}{2520}\right), 2\right), J \cdot \ell, U\right) \]

    15. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\left(\ell \cdot \color{blue}{{\ell}^{2}}\right) \cdot \frac{1}{2520}\right), 2\right), J \cdot \ell, U\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\color{blue}{\left(\ell \cdot {\ell}^{2}\right)} \cdot \frac{1}{2520}\right), 2\right), J \cdot \ell, U\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{1}{2520}\right), 2\right), J \cdot \ell, U\right) \]

    18. *-lowering-*.f64 79.1

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot 0.0003968253968253968\right), 2\right), J \cdot \ell, U\right) \]

11. Simplified 79.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot 0.0003968253968253968\right)}, 2\right), J \cdot \ell, U\right) \]

12. Final simplification 79.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \left(0.0003968253968253968 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right), 2\right), \ell \cdot J, U\right) \]
13. Add Preprocessing

Alternative 18: 73.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), \ell \cdot J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  (fma (* l l) (fma l (* l 0.016666666666666666) 0.3333333333333333) 2.0)
  (* l J)
  U))

C

double code(double J, double l, double K, double U) {
	return fma(fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0), (l * J), U);
}

Julia

function code(J, l, K, U)
	return fma(fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0), Float64(l * J), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * J), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. +-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. *-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. *-lowering-*.f64 96.2

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

5. Simplified 96.2%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right), J \cdot \ell, U\right)} \]

8. Simplified 79.3%

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

9. Taylor expanded in l around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}, J \cdot \ell, U\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2}, J \cdot \ell, U\right) \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right)}, J \cdot \ell, U\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right), J \cdot \ell, U\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, 2\right), J \cdot \ell, U\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{1}{60} \cdot {\ell}^{2} + \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \frac{1}{60}} + \frac{1}{3}, 2\right), J \cdot \ell, U\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60} + \frac{1}{3}, 2\right), J \cdot \ell, U\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{60}\right)} + \frac{1}{3}, 2\right), J \cdot \ell, U\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \color{blue}{\left(\frac{1}{60} \cdot \ell\right)} + \frac{1}{3}, 2\right), J \cdot \ell, U\right) \]

    10. accelerator-lowering-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{60} \cdot \ell, \frac{1}{3}\right)}, 2\right), J \cdot \ell, U\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{60}}, \frac{1}{3}\right), 2\right), J \cdot \ell, U\right) \]

    12. *-lowering-*.f64 77.0

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.016666666666666666}, 0.3333333333333333\right), 2\right), J \cdot \ell, U\right) \]

11. Simplified 77.0%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)}, J \cdot \ell, U\right) \]

12. Final simplification 77.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), \ell \cdot J, U\right) \]
13. Add Preprocessing

Alternative 19: 70.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 0.3333333333333333 (* l (* l l))))))
   (if (<= l -2.2e+27) t_0 (if (<= l 3.5e+71) (fma 2.0 (* l J) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (0.3333333333333333 * (l * (l * l)));
	double tmp;
	if (l <= -2.2e+27) {
		tmp = t_0;
	} else if (l <= 3.5e+71) {
		tmp = fma(2.0, (l * J), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(0.3333333333333333 * Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -2.2e+27)
		tmp = t_0;
	elseif (l <= 3.5e+71)
		tmp = fma(2.0, Float64(l * J), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.2e+27], t$95$0, If[LessEqual[l, 3.5e+71], N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1999999999999999e27 or 3.4999999999999999e71 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      13. *-lowering-*.f64 60.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \frac{1}{3}\right)} \cdot {\ell}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{{\ell}^{2}} \cdot \ell\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \]

       8. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right) \]

       10. unpow3 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{{\ell}^{3}}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       12. cube-mult N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right) \]

       15. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

       16. *-lowering-*.f64 65.8

           \[\leadsto J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   11. Simplified 65.8%

       \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -2.1999999999999999e27 < l < 3.4999999999999999e71

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.0

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

   5. Simplified 91.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

      13. *-lowering-*.f64 75.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. *-lowering-*.f64 75.6

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

   11. Simplified 75.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 68.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \ell \cdot J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (fma l (* l 0.3333333333333333) 2.0) (* l J) U))

C

double code(double J, double l, double K, double U) {
	return fma(fma(l, (l * 0.3333333333333333), 2.0), (l * J), U);
}

Julia

function code(J, l, K, U)
	return fma(fma(l, Float64(l * 0.3333333333333333), 2.0), Float64(l * J), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * J), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 86.9

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

5. Simplified 86.9%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

   13. *-lowering-*.f64 70.3

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

8. Simplified 70.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

9. Final simplification 70.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \ell \cdot J, U\right) \]
10. Add Preprocessing

Alternative 21: 53.2% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in l around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 86.9

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

5. Simplified 86.9%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right)} + U \]

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2, J \cdot \ell, U\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2, J \cdot \ell, U\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \ell\right)} + 2, J \cdot \ell, U\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right)}, J \cdot \ell, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2\right), J \cdot \ell, U\right) \]

   13. *-lowering-*.f64 70.3

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{J \cdot \ell}, U\right) \]

8. Simplified 70.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J \cdot \ell, U\right)} \]

9. Taylor expanded in l around 0

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

    1. +-commutative N/A

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

    2. accelerator-lowering-fma.f64 N/A

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

    3. *-lowering-*.f64 58.1

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

11. Simplified 58.1%

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

12. Final simplification 58.1%

    \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, U\right) \]
13. Add Preprocessing

Alternative 22: 36.2% accurate, 330.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in J around 0

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

5. Simplified 39.2%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))