Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 13.8s

Alternatives: 20

Speedup: 2.1×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% 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 86.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. flip-- N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. associate-*r/ N/A

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

   9. flip-- N/A

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

   10. lift--.f64 N/A

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

   11. associate-*l* N/A

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

4. Applied rewrites 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. Add Preprocessing

Alternative 2: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (*
     (cos (* K 0.5))
     (*
      2.0
      (fma
       (fma
        (* l l)
        (fma (* l l) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       (* l (* l l))
       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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (2.0 * fma(fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (l * (l * l)), 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * fma(fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(l * Float64(l * l)), 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   1. Initial program 86.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (*
     (cos (* K 0.5))
     (*
      l
      (fma
       (* l l)
       (fma
        (* l l)
        (fma l (* l 0.0003968253968253968) 0.016666666666666666)
        0.3333333333333333)
       2.0)))
    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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (l * fma((l * l), fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0))), 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0))), 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   1. Initial program 86.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (*
     (cos (* K 0.5))
     (* l (fma (* l l) (* (* l l) (* l (* l 0.0003968253968253968))) 2.0)))
    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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (l * fma((l * l), ((l * l) * (l * (l * 0.0003968253968253968))), 2.0))), 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(Float64(l * l), Float64(Float64(l * l) * Float64(l * Float64(l * 0.0003968253968253968))), 2.0))), 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   1. Initial program 86.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 97.6

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

   10. Applied rewrites 97.6%

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

   11. Taylor expanded in l around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 97.4

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

   13. Applied rewrites 97.4%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (*
     (cos (* K 0.5))
     (*
      2.0
      (fma
       (fma (* l l) 0.008333333333333333 0.16666666666666666)
       (* l (* l l))
       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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (2.0 * fma(fma((l * l), 0.008333333333333333, 0.16666666666666666), (l * (l * l)), 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * fma(fma(Float64(l * l), 0.008333333333333333, 0.16666666666666666), Float64(l * Float64(l * l)), 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   1. Initial program 86.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{1}{120}}, \frac{1}{6}\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{0.008333333333333333}, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (* (cos (* K 0.5)) (* 2.0 (fma l (* l (* l 0.16666666666666666)) 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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (2.0 * fma(l, (l * (l * 0.16666666666666666)), 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * fma(l, Float64(l * Float64(l * 0.16666666666666666)), 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(l * N[(l * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   1. Initial program 86.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{6} \cdot {\ell}^{2} + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{6} \cdot {\ell}^{2}\right) + \ell \cdot 1\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \left(\frac{1}{6} \cdot {\ell}^{2}\right) + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{6} \cdot {\ell}^{2}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\ell, \color{blue}{{\ell}^{2} \cdot \frac{1}{6}}, \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 91.9

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)}, \ell\right)\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   7. Applied rewrites 91.9%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(\cos \left(K \cdot 0.5\right) \cdot J\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma l (* (* (cos (* K 0.5)) J) (fma l (* l 0.3333333333333333) 2.0)) 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)) <= 0.97) {
		tmp = fma(l, ((cos((K * 0.5)) * J) * fma(l, (l * 0.3333333333333333), 2.0)), 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)) <= 0.97)
		tmp = fma(l, Float64(Float64(cos(Float64(K * 0.5)) * J) * fma(l, Float64(l * 0.3333333333333333), 2.0)), 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], 0.97], N[(l * N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $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))) < 0.96999999999999997

   1. Initial program 86.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-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. Applied rewrites 90.4%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 99.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(\cos \left(K \cdot 0.5\right) \cdot J\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (fma -0.125 (* K K) 1.0)
    (*
     J
     (*
      l
      (fma (* l l) (fma (* l l) 0.016666666666666666 0.3333333333333333) 2.0)))
    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(fma(-0.125, (K * K), 1.0), (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), 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(fma(-0.125, Float64(K * K), 1.0), Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), 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[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $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] + 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 95.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)} \]

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

   1. Initial program 83.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 93.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U, \left(J \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right)\right) \cdot \frac{2}{U}, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (fma -0.125 (* K K) 1.0)
    (*
     J
     (*
      l
      (fma (* l l) (fma (* l l) 0.016666666666666666 0.3333333333333333) 2.0)))
    U)
   (fma
    U
    (*
     (*
      J
      (fma
       (* l (* l l))
       (fma
        l
        (* l (fma (* l l) 0.0001984126984126984 0.008333333333333333))
        0.16666666666666666)
       l))
     (/ 2.0 U))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(fma(-0.125, (K * K), 1.0), (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma(U, ((J * fma((l * (l * l)), fma(l, (l * fma((l * l), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), l)) * (2.0 / U)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(fma(-0.125, Float64(K * K), 1.0), Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(U, Float64(Float64(J * fma(Float64(l * Float64(l * l)), fma(l, Float64(l * fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), l)) * Float64(2.0 / U)), 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[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $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] + U), $MachinePrecision], N[(U * N[(N[(J * N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * N[(2.0 / U), $MachinePrecision]), $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 95.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)} \]

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

   1. Initial program 83.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 l around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \color{blue}{1}, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.0%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \color{blue}{1}, J, U\right) \]

   11. Taylor expanded in U around inf

       \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell + {\ell}^{3} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto U \cdot \color{blue}{\left(2 \cdot \frac{J \cdot \left(\ell + {\ell}^{3} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)}{U} + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{U \cdot \left(2 \cdot \frac{J \cdot \left(\ell + {\ell}^{3} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)}{U}\right) + U \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto U \cdot \left(2 \cdot \frac{J \cdot \left(\ell + {\ell}^{3} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)}{U}\right) + \color{blue}{U} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(U, 2 \cdot \frac{J \cdot \left(\ell + {\ell}^{3} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)}{U}, U\right)} \]

   13. Applied rewrites 90.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(U, \left(J \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right)\right) \cdot \frac{2}{U}, U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (fma -0.125 (* K K) 1.0)
    (*
     J
     (*
      l
      (fma (* l l) (fma (* l l) 0.016666666666666666 0.3333333333333333) 2.0)))
    U)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       l
       (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
       0.3333333333333333)
      2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(fma(-0.125, (K * K), 1.0), (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma((l * fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(fma(-0.125, Float64(K * K), 1.0), Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), 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[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $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] + U), $MachinePrecision], 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] * 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 95.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)} \]

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

   1. Initial program 83.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 l around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 95.5

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in K around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 89.0

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

   13. Applied rewrites 89.0%

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

Alternative 11: 82.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma -0.25 (* K K) 2.0)) J U)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       l
       (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
       0.3333333333333333)
      2.0))
    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(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma((l * fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), 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(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0)), 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[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], 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] * 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 95.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 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. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      6. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

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

   1. Initial program 83.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 l around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 95.5

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in K around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 89.0

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

   13. Applied rewrites 89.0%

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

Alternative 12: 81.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma -0.25 (* K K) 2.0)) J U)
   (fma
    (*
     2.0
     (fma
      (fma (* l l) 0.008333333333333333 0.16666666666666666)
      (* l (* l l))
      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(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma((2.0 * fma(fma((l * l), 0.008333333333333333, 0.16666666666666666), (l * (l * l)), 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(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(Float64(2.0 * fma(fma(Float64(l * l), 0.008333333333333333, 0.16666666666666666), Float64(l * Float64(l * l)), 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[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + 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 95.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 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. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      6. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

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

   1. Initial program 83.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 l around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell + 1 \cdot \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}\right)} \cdot \ell + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right) + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{{\ell}^{3}} + 1 \cdot \ell\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{3} + \color{blue}{\ell}\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{3}, \ell\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \color{blue}{1}, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.0%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot \color{blue}{1}, J, U\right) \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{1}{120}}, \frac{1}{6}\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot 1, J, U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 86.9%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{0.008333333333333333}, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)\right) \cdot 1, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left(\ell \cdot 0.3333333333333333\right) \cdot \left(t\_1 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -2.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(2 \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (* l 0.3333333333333333) (* t_1 (* J (* l l))))))
   (if (<= l -3.4e+140)
     t_2
     (if (<= l -2.9)
       t_0
       (if (<= l 1.14e-10)
         (fma (* t_1 (* 2.0 J)) l U)
         (if (<= l 4.2e+138) 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 = (l * 0.3333333333333333) * (t_1 * (J * (l * l)));
	double tmp;
	if (l <= -3.4e+140) {
		tmp = t_2;
	} else if (l <= -2.9) {
		tmp = t_0;
	} else if (l <= 1.14e-10) {
		tmp = fma((t_1 * (2.0 * J)), l, U);
	} else if (l <= 4.2e+138) {
		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(Float64(l * 0.3333333333333333) * Float64(t_1 * Float64(J * Float64(l * l))))
	tmp = 0.0
	if (l <= -3.4e+140)
		tmp = t_2;
	elseif (l <= -2.9)
		tmp = t_0;
	elseif (l <= 1.14e-10)
		tmp = fma(Float64(t_1 * Float64(2.0 * J)), l, U);
	elseif (l <= 4.2e+138)
		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[(N[(l * 0.3333333333333333), $MachinePrecision] * N[(t$95$1 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+140], t$95$2, If[LessEqual[l, -2.9], t$95$0, If[LessEqual[l, 1.14e-10], N[(N[(t$95$1 * N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[l, 4.2e+138], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4e140 or 4.20000000000000014e138 < 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

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. lower-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. Applied rewrites 100.0%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -3.4e140 < l < -2.89999999999999991 or 1.1399999999999999e-10 < l < 4.20000000000000014e138

   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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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. Applied rewrites 78.3%

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

   if -2.89999999999999991 < l < 1.1399999999999999e-10

   1. Initial program 71.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.7

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

      20. lift-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2.9:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), \ell \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma -0.25 (* K K) 2.0)) J U)
   (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) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * fma(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma(fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0), (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(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0), Float64(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[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * 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 95.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 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. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      6. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 86.9

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

   8. Applied rewrites 86.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), \ell \cdot J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma -0.25 (* K K) 2.0)) J U)
   (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) 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(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), 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(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), 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[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $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 95.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. associate-*l* N/A

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

   4. Applied rewrites 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 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. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      6. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. lower-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. Applied rewrites 85.4%

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

   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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.3

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

   8. Applied rewrites 83.3%

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

Alternative 16: 76.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l J) (fma -0.25 (* K K) 2.0) U)
   (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), 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 * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), 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 * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $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 95.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)} + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}} + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(J \cdot \ell\right) \cdot \frac{-1}{4}\right)} \cdot {K}^{2} + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      6. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(J \cdot \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)} + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(\left(J \cdot \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right) + \color{blue}{\left(J \cdot \ell\right) \cdot 2}\right) + U \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)} + U \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \frac{-1}{4} \cdot {K}^{2} + 2, U\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \ell}, \frac{-1}{4} \cdot {K}^{2} + 2, U\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(J \cdot \ell, \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, U\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), U\right) \]

      13. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. lower-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. Applied rewrites 85.4%

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

   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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.3

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

   8. Applied rewrites 83.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.4% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) U))

C

double code(double J, double l, double K, double U) {
	return fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), U);
}

Julia

function code(J, l, K, U)
	return fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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. lower-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. Applied rewrites 86.4%

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

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. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 71.5

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

8. Applied rewrites 71.5%

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

Alternative 18: 54.3% accurate, 27.5× speedup

Math

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

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(Float64(2.0 * l), J, U)
end

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. 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. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Applied rewrites 55.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 55.3

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

10. Applied rewrites 55.3%

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

Alternative 19: 54.3% 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 86.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. 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. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Applied rewrites 55.3%

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

9. Final simplification 55.3%

   \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, U\right) \]
10. Add Preprocessing

Alternative 20: 19.8% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. 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. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Applied rewrites 55.3%

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

9. Taylor expanded in J around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 22.2

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

11. Applied rewrites 22.2%

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

12. Final simplification 22.2%

    \[\leadsto 2 \cdot \left(\ell \cdot J\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))