Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 100.0%

Time: 13.2s

Alternatives: 22

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% 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 86.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sinh-undef N/A

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

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

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

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

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

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

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

   9. div-inv N/A

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

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

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

   11. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (*
        2.0
        (*
         J
         (fma
          l
          (*
           (* K K)
           (fma
            K
            (* K (fma K (* K -2.170138888888889e-5) 0.0026041666666666665))
            -0.125))
          l)))
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = 2.0 * (J * fma(l, ((K * K) * fma(K, (K * fma(K, (K * -2.170138888888889e-5), 0.0026041666666666665)), -0.125)), l));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(2.0 * Float64(J * fma(l, Float64(Float64(K * K) * fma(K, Float64(K * fma(K, Float64(K * -2.170138888888889e-5), 0.0026041666666666665)), -0.125)), l)));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(2.0 * N[(J * N[(l * N[(N[(K * K), $MachinePrecision] * N[(K * N[(K * N[(K * N[(K * -2.170138888888889e-5), $MachinePrecision] + 0.0026041666666666665), $MachinePrecision]), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 88.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 60.6

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

   5. Simplified 60.6%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 46.6

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

   8. Simplified 46.6%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 62.7

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

   10. Applied egg-rr 62.7%

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

   if -0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 83.8%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({K}^{2}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) + \color{blue}{\frac{-1}{8}}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right)}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{-1}{46080}} + \frac{1}{384}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{46080}, \frac{1}{384}\right)}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{46080}, \frac{1}{384}\right), \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      14. *-lowering-*.f64 63.7

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]

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

          \[\leadsto 2 \cdot \color{blue}{\left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + 1\right)}\right)\right) \]

       4. distribute-lft-in N/A

          \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right) + \ell \cdot 1\right)}\right) \]

       5. *-rgt-identity N/A

          \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right) + \color{blue}{\ell}\right)\right) \]

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

          \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left(\ell, {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right), \ell\right)}\right) \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)} \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 94.6

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (*
        2.0
        (*
         J
         (fma
          l
          (*
           (* K K)
           (fma
            K
            (* K (fma K (* K -2.170138888888889e-5) 0.0026041666666666665))
            -0.125))
          l)))
       (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 t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = 2.0 * (J * fma(l, ((K * K) * fma(K, (K * fma(K, (K * -2.170138888888889e-5), 0.0026041666666666665)), -0.125)), l));
	} 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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(2.0 * Float64(J * fma(l, Float64(Float64(K * K) * fma(K, Float64(K * fma(K, Float64(K * -2.170138888888889e-5), 0.0026041666666666665)), -0.125)), l)));
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(2.0 * N[(J * N[(l * N[(N[(K * K), $MachinePrecision] * N[(K * N[(K * N[(K * N[(K * -2.170138888888889e-5), $MachinePrecision] + 0.0026041666666666665), $MachinePrecision]), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 88.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 60.6

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

   5. Simplified 60.6%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 46.6

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

   8. Simplified 46.6%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 62.7

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

   10. Applied egg-rr 62.7%

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

   if -0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 83.8%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({K}^{2}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) + \color{blue}{\frac{-1}{8}}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right)}, 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

         \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{-1}{46080}} + \frac{1}{384}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

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

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{46080}, \frac{1}{384}\right)}, \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{46080}, \frac{1}{384}\right), \frac{-1}{8}\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

      14. *-lowering-*.f64 63.7

          \[\leadsto \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), 1\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), 1\right)} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right) + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]

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

          \[\leadsto 2 \cdot \color{blue}{\left(J \cdot \left(\ell \cdot \left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + 1\right)}\right)\right) \]

       4. distribute-lft-in N/A

          \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right) + \ell \cdot 1\right)}\right) \]

       5. *-rgt-identity N/A

          \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right) + \color{blue}{\ell}\right)\right) \]

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

          \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left(\ell, {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right), \ell\right)}\right) \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)} \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around 0

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

      1. *-lowering-*.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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

   10. Simplified 89.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;2 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \left(K \cdot K\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right), -0.125\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.996:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.996)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma
           (* l l)
           (fma (* l l) 0.0003968253968253968 0.016666666666666666)
           0.3333333333333333)
          2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.996) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.996)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.996], N[(U + N[(t$95$0 * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.996

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 94.2

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

   5. Simplified 94.2%

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

   if 0.996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 100.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.996:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\ell \cdot \mathsf{fma}\left(J, \left(K \cdot K\right) \cdot -0.125, J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (*
        (fma l (* l 0.3333333333333333) 2.0)
        (* l (fma J (* (* K K) -0.125) J)))
       (fma
        (*
         l
         (fma
          l
          (* l (fma (* l l) 0.016666666666666666 0.3333333333333333))
          2.0))
        J
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = fma(l, (l * 0.3333333333333333), 2.0) * (l * fma(J, ((K * K) * -0.125), J));
	} else {
		tmp = fma((l * fma(l, (l * fma((l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * Float64(l * fma(J, Float64(Float64(K * K) * -0.125), J)));
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 88.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 60.6

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

   5. Simplified 60.6%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 46.6

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

   8. Simplified 46.6%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 62.7

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

   10. Applied egg-rr 62.7%

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

   if -0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 83.8%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.8

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

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

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

   8. Simplified 60.3%

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

   9. Taylor expanded in U around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

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

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

       14. +-commutative N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. associate-*r* N/A

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

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

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

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

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

       20. unpow2 N/A

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

       21. *-lowering-*.f64 60.3

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

   11. Simplified 60.3%

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

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 87.6

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

   10. Simplified 87.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.8:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\ell \cdot \mathsf{fma}\left(J, \left(K \cdot K\right) \cdot -0.125, J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.69:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot \left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.69)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (* (* (* K K) -0.125) (* (* l J) (fma l (* l 0.3333333333333333) 2.0)))
       (fma
        (*
         l
         (fma
          l
          (* l (fma (* l l) 0.016666666666666666 0.3333333333333333))
          2.0))
        J
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.69) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = ((K * K) * -0.125) * ((l * J) * fma(l, (l * 0.3333333333333333), 2.0));
	} else {
		tmp = fma((l * fma(l, (l * fma((l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.69)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(Float64(Float64(K * K) * -0.125) * Float64(Float64(l * J) * fma(l, Float64(l * 0.3333333333333333), 2.0)));
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.69], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(l * J), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.68999999999999995

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 61.0

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

   5. Simplified 61.0%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 43.0

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

   8. Simplified 43.0%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 55.1

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

   10. Applied egg-rr 55.1%

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

   if -0.68999999999999995 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 79.4%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

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

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

   8. Simplified 79.4%

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

   9. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

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

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

       7. associate-*r* N/A

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

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

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 79.4

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

   11. Simplified 79.4%

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

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 87.6

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

   10. Simplified 87.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.69:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot \left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.996:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.996)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma (* l l) 0.016666666666666666 0.3333333333333333)
          2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.996) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.996)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.996], N[(U + N[(t$95$0 * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 0.996

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 93.3

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

   5. Simplified 93.3%

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

   if 0.996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 100.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.996:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.82)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (* J (* l (* -0.25 (* K K))))
       (fma
        (*
         l
         (fma
          l
          (* l (fma (* l l) 0.016666666666666666 0.3333333333333333))
          2.0))
        J
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.82) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = J * (l * (-0.25 * (K * K)));
	} else {
		tmp = fma((l * fma(l, (l * fma((l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.82)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(J * Float64(l * Float64(-0.25 * Float64(K * K))));
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.82], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(J * N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 59.2

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

   5. Simplified 59.2%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 64.8

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

   10. Applied egg-rr 64.8%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 62.0

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

   5. Simplified 62.0%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 50.2

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

   8. Simplified 50.2%

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

   9. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 54.2

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

   11. Simplified 54.2%

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

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 87.6

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

   10. Simplified 87.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, 2 \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.82)
     (fma (* (* l J) (* K -0.25)) K (fma 2.0 (* l J) U))
     (if (<= t_0 -0.3225)
       (* J (* l (* -0.25 (* K K))))
       (fma (fma (* l l) (* l 0.3333333333333333) (* 2.0 l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.82) {
		tmp = fma(((l * J) * (K * -0.25)), K, fma(2.0, (l * J), U));
	} else if (t_0 <= -0.3225) {
		tmp = J * (l * (-0.25 * (K * K)));
	} else {
		tmp = fma(fma((l * l), (l * 0.3333333333333333), (2.0 * l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.82)
		tmp = fma(Float64(Float64(l * J) * Float64(K * -0.25)), K, fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.3225)
		tmp = Float64(J * Float64(l * Float64(-0.25 * Float64(K * K))));
	else
		tmp = fma(fma(Float64(l * l), Float64(l * 0.3333333333333333), Float64(2.0 * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.82], N[(N[(N[(l * J), $MachinePrecision] * N[(K * -0.25), $MachinePrecision]), $MachinePrecision] * K + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.3225], N[(J * N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 59.2

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

   5. Simplified 59.2%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 64.8

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

   10. Applied egg-rr 64.8%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 62.0

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

   5. Simplified 62.0%

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

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      11. accelerator-lowering-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. *-lowering-*.f64 50.2

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

   8. Simplified 50.2%

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

   9. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 54.2

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

   11. Simplified 54.2%

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

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 82.0

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

   10. Simplified 82.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, J, U\right) \]
   11. Step-by-step derivation

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 82.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, \color{blue}{\ell \cdot 2}\right), J, U\right) \]

   12. Applied egg-rr 82.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, \ell \cdot 2\right)}, J, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(K \cdot -0.25\right), K, \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, 2 \cdot \ell\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.05)
     (+ U (* t_0 (* J (* l (fma 0.3333333333333333 (* l l) 2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.05) {
		tmp = U + (t_0 * (J * (l * fma(0.3333333333333333, (l * l), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], N[(U + N[(t$95$0 * N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.050000000000000003

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 92.4

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 92.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sinh \ell}, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      4. sinh-lowering-sinh.f64 96.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell} \cdot 2, J, U\right) \]

   9. Applied egg-rr 96.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.05)
   (fma l (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (* l l) 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)) <= 0.05) {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(0.3333333333333333, (l * l), 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)) <= 0.05)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, Float64(l * l), 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], 0.05], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[(l * l), $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))) < 0.050000000000000003

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), U\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)} \]

   if 0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sinh \ell}, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      4. sinh-lowering-sinh.f64 96.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell} \cdot 2, J, U\right) \]

   9. Applied egg-rr 96.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.05)
   (fma (* (cos (* K 0.5)) (* 2.0 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)) <= 0.05) {
		tmp = fma((cos((K * 0.5)) * (2.0 * 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)) <= 0.05)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * 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], 0.05], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $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))) < 0.050000000000000003

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 65.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   7. Simplified 65.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   if 0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sinh \ell}, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      4. sinh-lowering-sinh.f64 96.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell} \cdot 2, J, U\right) \]

   9. Applied egg-rr 96.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.05)
   (fma (cos (* K 0.5)) (* J (* 2.0 l)) 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)) <= 0.05) {
		tmp = fma(cos((K * 0.5)), (J * (2.0 * l)), 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)) <= 0.05)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(2.0 * l)), 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], 0.05], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $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))) < 0.050000000000000003

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. accelerator-lowering-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. cos-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   if 0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sinh \ell}, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]

      4. sinh-lowering-sinh.f64 96.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell} \cdot 2, J, U\right) \]

   9. Applied egg-rr 96.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sinh \ell \cdot 2}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3225)
   (fma (* l (fma 0.3333333333333333 (* l l) 2.0)) (* J (* (* K K) -0.125)) 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)) <= -0.3225) {
		tmp = fma((l * fma(0.3333333333333333, (l * l), 2.0)), (J * ((K * K) * -0.125)), 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)) <= -0.3225)
		tmp = fma(Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), Float64(J * Float64(Float64(K * K) * -0.125)), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * 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], -0.3225], N[(N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $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))) < -0.32250000000000001

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 90.9

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 90.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(J \cdot \color{blue}{\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)}\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot {K}^{2}} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left({K}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right)\right) \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot \left({K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right) + J\right)} + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right), {K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right) + J, U\right)} \]

   8. Simplified 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)}, U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}}, U\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\left(J \cdot \frac{-1}{8}\right)} \cdot {K}^{2}, U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), J \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), J \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(K \cdot K\right)}\right), U\right) \]

       7. *-lowering-*.f64 54.6

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right), U\right) \]

   11. Simplified 54.6%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)}, U\right) \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.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. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + 2\right), J, U\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right), 2\right)}, J, U\right) \]

   10. Simplified 89.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3225)
   (fma (* l (fma 0.3333333333333333 (* l l) 2.0)) (* J (* (* K K) -0.125)) U)
   (fma
    (*
     l
     (fma l (* l (fma (* l l) 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)) <= -0.3225) {
		tmp = fma((l * fma(0.3333333333333333, (l * l), 2.0)), (J * ((K * K) * -0.125)), U);
	} else {
		tmp = fma((l * fma(l, (l * fma((l * l), 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)) <= -0.3225)
		tmp = fma(Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), Float64(J * Float64(Float64(K * K) * -0.125)), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 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], -0.3225], N[(N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $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))) < -0.32250000000000001

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 90.9

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 90.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(J \cdot \color{blue}{\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)}\right) + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot {K}^{2}} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left({K}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right)\right) \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + U \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot \left({K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right) + J\right)} + U \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right), {K}^{2} \cdot \left(\frac{-1}{8} \cdot J\right) + J, U\right)} \]

   8. Simplified 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)}, U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot {K}^{2}}, U\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{\left(J \cdot \frac{-1}{8}\right)} \cdot {K}^{2}, U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), J \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right)}, U\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), J \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(K \cdot K\right)}\right), U\right) \]

       7. *-lowering-*.f64 54.6

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right), U\right) \]

   11. Simplified 54.6%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \color{blue}{J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)}, U\right) \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}, J, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2\right)}, J, U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2\right), J, U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} + 2\right), J, U\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right), 2\right)}, J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}, 2\right), J, U\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2} + \frac{1}{3}\right)}, 2\right), J, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{60}} + \frac{1}{3}\right), 2\right), J, U\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{60}, \frac{1}{3}\right)}, 2\right), J, U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{60}, \frac{1}{3}\right), 2\right), J, U\right) \]

      11. *-lowering-*.f64 87.6

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right) \]

   10. Simplified 87.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, 2 \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3225)
   (* J (* l (* -0.25 (* K K))))
   (fma (fma (* l l) (* l 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)) <= -0.3225) {
		tmp = J * (l * (-0.25 * (K * K)));
	} else {
		tmp = fma(fma((l * l), (l * 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)) <= -0.3225)
		tmp = Float64(J * Float64(l * Float64(-0.25 * Float64(K * K))));
	else
		tmp = fma(fma(Float64(l * l), Float64(l * 0.3333333333333333), Float64(2.0 * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.3225], N[(J * N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 60.5

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \frac{-1}{4} \cdot {K}^{2} + 2, U\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \ell}, \frac{-1}{4} \cdot {K}^{2} + 2, U\right) \]

      11. accelerator-lowering-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. *-lowering-*.f64 49.1

          \[\leadsto \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right), U\right) \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \frac{-1}{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       8. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto J \cdot \left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

       12. *-lowering-*.f64 49.1

           \[\leadsto J \cdot \left(\ell \cdot \left(-0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 49.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)} \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.6%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, J, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}, J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right), J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right), J, U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right), J, U\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right)}, J, U\right) \]

      7. *-lowering-*.f64 82.0

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right), J, U\right) \]

   10. Simplified 82.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, J, U\right) \]
   11. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right) + \ell \cdot 2}, J, U\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \frac{1}{3}\right)} + \ell \cdot 2, J, U\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \ell}, J, U\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \frac{1}{3}, 2 \cdot \ell\right)}, J, U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \ell \cdot \frac{1}{3}, 2 \cdot \ell\right), J, U\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \frac{1}{3}}, 2 \cdot \ell\right), J, U\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \frac{1}{3}, \color{blue}{\ell \cdot 2}\right), J, U\right) \]

       8. *-lowering-*.f64 82.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, \color{blue}{\ell \cdot 2}\right), J, U\right) \]

   12. Applied egg-rr 82.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, \ell \cdot 2\right)}, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.3333333333333333, 2 \cdot \ell\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3225)
   (* J (* l (* -0.25 (* K K))))
   (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)) <= -0.3225) {
		tmp = J * (l * (-0.25 * (K * K)));
	} 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)) <= -0.3225)
		tmp = Float64(J * Float64(l * Float64(-0.25 * Float64(K * K))));
	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], -0.3225], N[(J * N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -0.32250000000000001

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 60.5

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \frac{-1}{4} \cdot {K}^{2} + 2, U\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \ell}, \frac{-1}{4} \cdot {K}^{2} + 2, U\right) \]

      11. accelerator-lowering-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. *-lowering-*.f64 49.1

          \[\leadsto \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right), U\right) \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \frac{-1}{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       8. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto J \cdot \left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

       12. *-lowering-*.f64 49.1

           \[\leadsto J \cdot \left(\ell \cdot \left(-0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 49.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)} \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. *-lowering-*.f64 87.4

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 87.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + U} \]

      2. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 82.0

         \[\leadsto \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right), U\right) \]

   8. Simplified 82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3225)
   (* J (* l (* -0.25 (* K K))))
   (fma 2.0 (* l J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.3225) {
		tmp = J * (l * (-0.25 * (K * K)));
	} else {
		tmp = fma(2.0, (l * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.3225)
		tmp = Float64(J * Float64(l * Float64(-0.25 * Float64(K * K))));
	else
		tmp = fma(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], -0.3225], N[(J * N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.32250000000000001

   1. Initial program 86.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 60.5

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \frac{-1}{4} \cdot {K}^{2} + 2, U\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \ell}, \frac{-1}{4} \cdot {K}^{2} + 2, U\right) \]

      11. accelerator-lowering-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. *-lowering-*.f64 49.1

          \[\leadsto \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right), U\right) \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \frac{-1}{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell\right)} \]

       8. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto J \cdot \left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

       12. *-lowering-*.f64 49.1

           \[\leadsto J \cdot \left(\ell \cdot \left(-0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 49.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)} \]

   if -0.32250000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 65.5

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

      3. *-lowering-*.f64 60.6

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3225:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -9500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (* (* l l) 0.3333333333333333)))))
   (if (<= l -9500000.0) t_0 (if (<= l 2e+23) (fma 2.0 (* l J) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -9500000.0) {
		tmp = t_0;
	} else if (l <= 2e+23) {
		tmp = fma(2.0, (l * J), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -9500000.0)
		tmp = t_0;
	elseif (l <= 2e+23)
		tmp = fma(2.0, Float64(l * J), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9500000.0], t$95$0, If[LessEqual[l, 2e+23], N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.5e6 or 1.9999999999999998e23 < 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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 74.2%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, J, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}, J, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right), J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right), J, U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right), J, U\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right)}, J, U\right) \]

      7. *-lowering-*.f64 58.5

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right), J, U\right) \]

   10. Simplified 58.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, J, U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{3}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left({\ell}^{3} \cdot \frac{1}{3}\right)} \]

       3. cube-mult N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)} \cdot \frac{1}{3}\right) \]

       4. unpow2 N/A

          \[\leadsto J \cdot \left(\left(\ell \cdot \color{blue}{{\ell}^{2}}\right) \cdot \frac{1}{3}\right) \]

       5. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto J \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

       11. *-lowering-*.f64 58.5

           \[\leadsto J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   13. Simplified 58.5%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -9.5e6 < l < 1.9999999999999998e23

   1. Initial program 74.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 93.8

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 93.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

      3. *-lowering-*.f64 82.2

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9500000:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-44}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* l J))))
   (if (<= l -2.1e-6) t_0 (if (<= l 2.8e-44) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -2.1e-6) {
		tmp = t_0;
	} else if (l <= 2.8e-44) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (l * j)
    if (l <= (-2.1d-6)) then
        tmp = t_0
    else if (l <= 2.8d-44) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -2.1e-6) {
		tmp = t_0;
	} else if (l <= 2.8e-44) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (l * J)
	tmp = 0
	if l <= -2.1e-6:
		tmp = t_0
	elif l <= 2.8e-44:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(l * J))
	tmp = 0.0
	if (l <= -2.1e-6)
		tmp = t_0;
	elseif (l <= 2.8e-44)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (l * J);
	tmp = 0.0;
	if (l <= -2.1e-6)
		tmp = t_0;
	elseif (l <= 2.8e-44)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.1e-6], t$95$0, If[LessEqual[l, 2.8e-44], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.0999999999999998e-6 or 2.8e-44 < l

   1. Initial program 96.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

      7. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

      11. *-lowering-*.f64 33.6

          \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Simplified 33.6%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

      3. *-lowering-*.f64 24.5

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right) \]

   8. Simplified 24.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]

       2. *-commutative N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot J\right)} \]

       3. *-lowering-*.f64 23.0

          \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot J\right)} \]

   11. Simplified 23.0%

       \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} \]

   if -2.0999999999999998e-6 < l < 2.8e-44

   1. Initial program 75.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 74.9%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 54.5% 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 86.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

   4. *-lowering-*.f64 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. cos-lowering-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(\left(2 \cdot J\right) \cdot \ell\right) + U \]

   7. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} + U \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   11. *-lowering-*.f64 64.5

       \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

5. Simplified 64.5%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)} + U \]

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. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

   3. *-lowering-*.f64 54.5

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right) \]

8. Simplified 54.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)} \]

9. Final simplification 54.5%

   \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, U\right) \]
10. Add Preprocessing

Alternative 22: 36.4% accurate, 330.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 86.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 36.8%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))