Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.9%

Time: 14.1s

Alternatives: 23

Speedup: 2.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. *-commutative N/A

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

4. Applied egg-rr 100.0%

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

Alternative 2: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= t_0 (- INFINITY))
     (fma (* J (sinh l)) 2.0 U)
     (if (<= t_0 5e-11)
       (fma (cos (* K 0.5)) (* J (* l 2.0)) U)
       (fma (* (sinh l) (fma (* K K) (* J -0.125) J)) 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((J * sinh(l)), 2.0, U);
	} else if (t_0 <= 5e-11) {
		tmp = fma(cos((K * 0.5)), (J * (l * 2.0)), U);
	} else {
		tmp = fma((sinh(l) * fma((K * K), (J * -0.125), J)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	elseif (t_0 <= 5e-11)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(l * 2.0)), U);
	else
		tmp = fma(Float64(sinh(l) * fma(Float64(K * K), Float64(J * -0.125), J)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[t$95$0, 5e-11], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(J * -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 79.4%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.00000000000000018e-11

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if 5.00000000000000018e-11 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.4

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

   7. Simplified 79.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma
           l
           (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
           0.3333333333333333)
          2.0)))))
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		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((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, 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, 0.97], N[(U + N[(t$95$0 * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.9

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

   5. Simplified 96.9%

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

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

   1. Initial program 91.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 99.5%

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

4. Final simplification 98.2%

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

Alternative 4: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma (* l l) 0.016666666666666666 0.3333333333333333)
          2.0)))))
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		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(J * sinh(l)), 2.0, 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, 0.97], 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[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 93.1

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

   5. Simplified 93.1%

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

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

   1. Initial program 91.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 99.5%

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

4. Final simplification 96.4%

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

Alternative 5: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \frac{\left(\ell \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right) \cdot \cos \left(K \cdot 0.5\right)}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    U
    (/ (* (* l (* J (fma 0.3333333333333333 (* l l) 2.0))) (cos (* K 0.5))) U)
    U)
   (fma (* J (sinh l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma(U, (((l * (J * fma(0.3333333333333333, (l * l), 2.0))) * cos((K * 0.5))) / U), U);
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(U, Float64(Float64(Float64(l * Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0))) * cos(Float64(K * 0.5))) / U), U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U * N[(N[(N[(l * N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in U around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Simplified 92.7%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 95.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \frac{\left(\ell \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right) \cdot \cos \left(K \cdot 0.5\right)}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.04)
     (+ U (* t_0 (* J (* l (fma 0.3333333333333333 (* l l) 2.0)))))
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = U + (t_0 * (J * (l * fma(0.3333333333333333, (l * l), 2.0))));
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)))));
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, 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, -0.04], 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[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-*.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. lower-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. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 95.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma l (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (cos (* K 0.5))) U)
   (fma (* J (sinh l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma(l, ((J * fma(0.3333333333333333, (l * l), 2.0)) * cos((K * 0.5))), U);
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(l, Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * cos(Float64(K * 0.5))), U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 88.5%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 95.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \left(\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right) \cdot \left(J \cdot \left(K \cdot K\right)\right)\right) \cdot \frac{-0.125}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    U
    (*
     (* (* l (fma l (* l 0.3333333333333333) 2.0)) (* J (* K K)))
     (/ -0.125 U))
    U)
   (fma (* J (sinh l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma(U, (((l * fma(l, (l * 0.3333333333333333), 2.0)) * (J * (K * K))) * (-0.125 / U)), U);
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(U, Float64(Float64(Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)) * Float64(J * Float64(K * K))) * Float64(-0.125 / U)), U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U * N[(N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / U), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in U around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Simplified 92.7%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. associate-*l* N/A

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

       6. distribute-lft-out N/A

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

       7. lower-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

   11. Simplified 26.6%

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

   12. Taylor expanded in K around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

   14. Simplified 60.9%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 95.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \left(\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right) \cdot \left(J \cdot \left(K \cdot K\right)\right)\right) \cdot \frac{-0.125}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \left(\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right) \cdot \left(J \cdot \left(K \cdot K\right)\right)\right) \cdot \frac{-0.125}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    U
    (*
     (* (* l (fma l (* l 0.3333333333333333) 2.0)) (* J (* K K)))
     (/ -0.125 U))
    U)
   (+
    U
    (*
     (fma
      (* l l)
      (fma
       l
       (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
       0.3333333333333333)
      2.0)
     (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma(U, (((l * fma(l, (l * 0.3333333333333333), 2.0)) * (J * (K * K))) * (-0.125 / U)), U);
	} else {
		tmp = U + (fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0) * (J * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(U, Float64(Float64(Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)) * Float64(J * Float64(K * K))) * Float64(-0.125 / U)), U);
	else
		tmp = Float64(U + Float64(fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0) * Float64(J * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U * N[(N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / U), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in U around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Simplified 92.7%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. associate-*l* N/A

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

       6. distribute-lft-out N/A

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

       7. lower-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

   11. Simplified 26.6%

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

   12. Taylor expanded in K around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

   14. Simplified 60.9%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(U, \left(\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right) \cdot \left(J \cdot \left(K \cdot K\right)\right)\right) \cdot \frac{-0.125}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (+
    U
    (*
     l
     (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma K (* K -0.125) 1.0))))
   (+
    U
    (*
     (fma
      (* l l)
      (fma
       l
       (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
       0.3333333333333333)
      2.0)
     (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = U + (l * ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(K, (K * -0.125), 1.0)));
	} else {
		tmp = U + (fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0) * (J * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = Float64(U + Float64(l * Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(K, Float64(K * -0.125), 1.0))));
	else
		tmp = Float64(U + Float64(fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0) * Float64(J * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U + N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.5

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

   8. Simplified 59.5%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (+
    U
    (*
     l
     (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma K (* K -0.125) 1.0))))
   (fma
    J
    (*
     l
     (fma
      (* l l)
      (fma
       l
       (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
       0.3333333333333333)
      2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = U + (l * ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(K, (K * -0.125), 1.0)));
	} else {
		tmp = fma(J, (l * fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = Float64(U + Float64(l * Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(K, Float64(K * -0.125), 1.0))));
	else
		tmp = fma(J, Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U + N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.5

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

   8. Simplified 59.5%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 89.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (+
    U
    (*
     l
     (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma K (* K -0.125) 1.0))))
   (fma
    (* l (fma (* l l) (* (* l l) (* (* l l) 0.0003968253968253968)) 2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = U + (l * ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(K, (K * -0.125), 1.0)));
	} else {
		tmp = fma((l * fma((l * l), ((l * l) * ((l * l) * 0.0003968253968253968)), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = Float64(U + Float64(l * Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(K, Float64(K * -0.125), 1.0))));
	else
		tmp = fma(Float64(l * fma(Float64(l * l), Float64(Float64(l * l) * Float64(Float64(l * l) * 0.0003968253968253968)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U + N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $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))) < -0.0400000000000000008

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.5

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

   8. Simplified 59.5%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   10. Applied egg-rr 89.7%

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

   11. Taylor expanded in l around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 89.7

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

   13. Simplified 89.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (+
    U
    (*
     l
     (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma K (* K -0.125) 1.0))))
   (fma
    J
    (*
     l
     (fma l (* l (fma l (* l 0.016666666666666666) 0.3333333333333333)) 2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = U + (l * ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(K, (K * -0.125), 1.0)));
	} else {
		tmp = fma(J, (l * fma(l, (l * fma(l, (l * 0.016666666666666666), 0.3333333333333333)), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = Float64(U + Float64(l * Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(K, Float64(K * -0.125), 1.0))));
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333)), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(U + N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 89.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 60.4

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

   8. Simplified 60.4%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 89.2%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 85.8

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

   11. Simplified 85.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(K \cdot \left(K \cdot \ell\right)\right) \cdot -0.25, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma J (* l 2.0) (fma J (* (* K (* K l)) -0.25) U))
   (fma
    J
    (*
     l
     (fma l (* l (fma l (* l 0.016666666666666666) 0.3333333333333333)) 2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(J, (l * 2.0), fma(J, ((K * (K * l)) * -0.25), U));
	} else {
		tmp = fma(J, (l * fma(l, (l * fma(l, (l * 0.016666666666666666), 0.3333333333333333)), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(J, Float64(l * 2.0), fma(J, Float64(Float64(K * Float64(K * l)) * -0.25), U));
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333)), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(J * N[(l * 2.0), $MachinePrecision] + N[(J * N[(N[(K * N[(K * l), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in K around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

   8. Simplified 51.8%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 51.8

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

   10. Applied egg-rr 51.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 89.2%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 85.8

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

   11. Simplified 85.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(K \cdot \left(K \cdot \ell\right)\right) \cdot -0.25, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(K \cdot \left(K \cdot \ell\right)\right) \cdot -0.25, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma J (* l 2.0) (fma J (* (* K (* K l)) -0.25) U))
   (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(J, (l * 2.0), fma(J, ((K * (K * l)) * -0.25), U));
	} else {
		tmp = fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(J, Float64(l * 2.0), fma(J, Float64(Float64(K * Float64(K * l)) * -0.25), U));
	else
		tmp = fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(J * N[(l * 2.0), $MachinePrecision] + N[(J * N[(N[(K * N[(K * l), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + U), $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))) < -0.050000000000000003

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in K around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

   8. Simplified 51.8%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 51.8

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

   10. Applied egg-rr 51.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 78.9%

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

   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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   8. Simplified 78.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(K \cdot \left(K \cdot \ell\right)\right) \cdot -0.25, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\ell, 2, \ell \cdot \left(\left(K \cdot K\right) \cdot -0.25\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma J (fma l 2.0 (* l (* (* K K) -0.25))) U)
   (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(J, fma(l, 2.0, (l * ((K * K) * -0.25))), U);
	} else {
		tmp = fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(J, fma(l, 2.0, Float64(l * Float64(Float64(K * K) * -0.25))), U);
	else
		tmp = fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(J * N[(l * 2.0 + N[(l * N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in K around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

   8. Simplified 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+r+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 51.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 51.8

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

   10. Applied egg-rr 51.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 78.9%

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

   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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   8. Simplified 78.2%

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

Alternative 17: 87.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \mathbf{if}\;\ell \leq -3.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* J (sinh l)) 2.0 U)))
   (if (<= l -3.9)
     t_0
     (if (<= l 2.55e-23)
       (fma (cos (* K 0.5)) (* J (* l 2.0)) U)
       (if (<= l 7e+104)
         t_0
         (*
          J
          (*
           (fma -0.125 (* K K) 1.0)
           (* l (fma l (* l 0.3333333333333333) 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((J * sinh(l)), 2.0, U);
	double tmp;
	if (l <= -3.9) {
		tmp = t_0;
	} else if (l <= 2.55e-23) {
		tmp = fma(cos((K * 0.5)), (J * (l * 2.0)), U);
	} else if (l <= 7e+104) {
		tmp = t_0;
	} else {
		tmp = J * (fma(-0.125, (K * K), 1.0) * (l * fma(l, (l * 0.3333333333333333), 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(J * sinh(l)), 2.0, U)
	tmp = 0.0
	if (l <= -3.9)
		tmp = t_0;
	elseif (l <= 2.55e-23)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(l * 2.0)), U);
	elseif (l <= 7e+104)
		tmp = t_0;
	else
		tmp = Float64(J * Float64(fma(-0.125, Float64(K * K), 1.0) * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]}, If[LessEqual[l, -3.9], t$95$0, If[LessEqual[l, 2.55e-23], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 7e+104], t$95$0, N[(J * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.89999999999999991 or 2.55000000000000005e-23 < l < 7.0000000000000003e104

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 80.6%

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

   if -3.89999999999999991 < l < 2.55000000000000005e-23

   1. Initial program 73.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if 7.0000000000000003e104 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   5. Simplified 86.1%

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

   6. Taylor expanded in U around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Simplified 90.7%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. associate-*l* N/A

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

       6. distribute-lft-out N/A

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

       7. lower-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

   11. Simplified 4.9%

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

   12. Taylor expanded in U around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

          \[\leadsto J \cdot \left(\mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right) \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto J \cdot \left(\mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right) \cdot \left(\ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto J \cdot \left(\mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right) \cdot \left(\ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto J \cdot \left(\mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right) \cdot \left(\ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right)\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto J \cdot \left(\mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right) \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right)}\right)\right) \]

       14. lower-*.f64 82.9

           \[\leadsto J \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right)\right)\right) \]

   14. Simplified 82.9%

       \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+37}:\\ \;\;\;\;J \cdot \mathsf{fma}\left(\ell, 2, \frac{U}{J}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4e+37)
   (* J (fma l 2.0 (/ U J)))
   (if (<= l 5.2e+34)
     (fma J (* l 2.0) U)
     (* l (* J (fma (* K K) -0.25 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4e+37) {
		tmp = J * fma(l, 2.0, (U / J));
	} else if (l <= 5.2e+34) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = l * (J * fma((K * K), -0.25, 2.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4e+37)
		tmp = Float64(J * fma(l, 2.0, Float64(U / J)));
	elseif (l <= 5.2e+34)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = Float64(l * Float64(J * fma(Float64(K * K), -0.25, 2.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4e+37], N[(J * N[(l * 2.0 + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+34], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.99999999999999982e37

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 25.1

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 25.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

      4. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

      7. lower-*.f64 23.2

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   8. Simplified 23.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

       2. *-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\ell \cdot 2} + \frac{U}{J}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto J \cdot \color{blue}{\mathsf{fma}\left(\ell, 2, \frac{U}{J}\right)} \]

       4. lower-/.f64 36.5

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, 2, \color{blue}{\frac{U}{J}}\right) \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{J \cdot \mathsf{fma}\left(\ell, 2, \frac{U}{J}\right)} \]

   if -3.99999999999999982e37 < l < 5.19999999999999995e34

   1. Initial program 76.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 90.1

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 90.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

      4. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

      7. lower-*.f64 75.4

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

   if 5.19999999999999995e34 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 27.9

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) + 2 \cdot \left(J \cdot \ell\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)}\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2} + U}\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\frac{-1}{4} \cdot \left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)} + U\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \frac{-1}{4} \cdot \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)} + U\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) + U\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4}} + U\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}\right)} + U\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\mathsf{fma}\left(J, \left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}, U\right)}\right) \]

   8. Simplified 40.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(\ell \cdot \left(K \cdot K\right)\right) \cdot -0.25, U\right)\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \left(2 \cdot \ell\right) \cdot J} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \color{blue}{2 \cdot \left(\ell \cdot J\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + 2 \cdot \color{blue}{\left(J \cdot \ell\right)} \]

       4. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \color{blue}{\left(2 \cdot J\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot J\right)} + \left(2 \cdot J\right) \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + \left(2 \cdot J\right) \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot {K}^{2}\right) \cdot \ell\right)} + \left(2 \cdot J\right) \cdot \ell \]

       8. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \ell} + \left(2 \cdot J\right) \cdot \ell \]

       9. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       11. *-commutative N/A

           \[\leadsto \ell \cdot \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}} + 2 \cdot J\right) \]

       12. associate-*l* N/A

           \[\leadsto \ell \cdot \left(\color{blue}{J \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)} + 2 \cdot J\right) \]

       13. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)} + 2 \cdot J\right) \]

       14. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right) + \color{blue}{J \cdot 2}\right) \]

       15. distribute-lft-out N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       16. lower-*.f64 N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       17. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right)\right) \]

       18. lower-fma.f64 N/A

           \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}\right) \]

       19. unpow2 N/A

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right)\right) \]

       20. lower-*.f64 40.4

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.25, 2\right)\right) \]

   11. Simplified 40.4%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 60.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right)\right)\\ \mathbf{if}\;\ell \leq -950:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J (fma (* K K) -0.25 2.0)))))
   (if (<= l -950.0) t_0 (if (<= l 5.2e+34) (fma J (* l 2.0) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * fma((K * K), -0.25, 2.0));
	double tmp;
	if (l <= -950.0) {
		tmp = t_0;
	} else if (l <= 5.2e+34) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * fma(Float64(K * K), -0.25, 2.0)))
	tmp = 0.0
	if (l <= -950.0)
		tmp = t_0;
	elseif (l <= 5.2e+34)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -950.0], t$95$0, If[LessEqual[l, 5.2e+34], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -950 or 5.19999999999999995e34 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 25.7

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 25.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) + 2 \cdot \left(J \cdot \ell\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + \left(U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U + \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)}\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2} + U}\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\frac{-1}{4} \cdot \left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)} + U\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \frac{-1}{4} \cdot \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)} + U\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) + U\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4}} + U\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}\right)} + U\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, \ell \cdot 2, \color{blue}{\mathsf{fma}\left(J, \left({K}^{2} \cdot \ell\right) \cdot \frac{-1}{4}, U\right)}\right) \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, \mathsf{fma}\left(J, \left(\ell \cdot \left(K \cdot K\right)\right) \cdot -0.25, U\right)\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \left(2 \cdot \ell\right) \cdot J} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \color{blue}{2 \cdot \left(\ell \cdot J\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + 2 \cdot \color{blue}{\left(J \cdot \ell\right)} \]

       4. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \color{blue}{\left(2 \cdot J\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot J\right)} + \left(2 \cdot J\right) \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + \left(2 \cdot J\right) \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot {K}^{2}\right) \cdot \ell\right)} + \left(2 \cdot J\right) \cdot \ell \]

       8. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \ell} + \left(2 \cdot J\right) \cdot \ell \]

       9. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       11. *-commutative N/A

           \[\leadsto \ell \cdot \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}} + 2 \cdot J\right) \]

       12. associate-*l* N/A

           \[\leadsto \ell \cdot \left(\color{blue}{J \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)} + 2 \cdot J\right) \]

       13. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)} + 2 \cdot J\right) \]

       14. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right) + \color{blue}{J \cdot 2}\right) \]

       15. distribute-lft-out N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       16. lower-*.f64 N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       17. *-commutative N/A

           \[\leadsto \ell \cdot \left(J \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right)\right) \]

       18. lower-fma.f64 N/A

           \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}\right) \]

       19. unpow2 N/A

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right)\right) \]

       20. lower-*.f64 34.8

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.25, 2\right)\right) \]

   11. Simplified 34.8%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right)\right)} \]

   if -950 < l < 5.19999999999999995e34

   1. Initial program 76.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

      4. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

      7. lower-*.f64 77.7

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   8. Simplified 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.4% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 74:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l 2.0))))
   (if (<= l -1.35e+38) t_0 (if (<= l 74.0) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= -1.35e+38) {
		tmp = t_0;
	} else if (l <= 74.0) {
		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 = j * (l * 2.0d0)
    if (l <= (-1.35d+38)) then
        tmp = t_0
    else if (l <= 74.0d0) 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 = J * (l * 2.0);
	double tmp;
	if (l <= -1.35e+38) {
		tmp = t_0;
	} else if (l <= 74.0) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * 2.0)
	tmp = 0
	if l <= -1.35e+38:
		tmp = t_0
	elif l <= 74.0:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (l <= -1.35e+38)
		tmp = t_0;
	elseif (l <= 74.0)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * 2.0);
	tmp = 0.0;
	if (l <= -1.35e+38)
		tmp = t_0;
	elseif (l <= 74.0)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+38], t$95$0, If[LessEqual[l, 74.0], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.34999999999999998e38 or 74 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 25.2

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 25.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

      4. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

      7. lower-*.f64 23.7

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   8. Simplified 23.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

       5. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

       6. lower-*.f64 23.4

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   11. Simplified 23.4%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   if -1.34999999999999998e38 < l < 74

   1. Initial program 75.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

      12. lower-*.f64 95.3

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

      4. *-commutative N/A

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

      7. lower-*.f64 79.8

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   8. Simplified 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

   9. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)}\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \color{blue}{\left(-2 \cdot \frac{J \cdot \ell}{U} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       5. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \left(-2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \left(\color{blue}{\left(-2 \cdot J\right) \cdot \frac{\ell}{U}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \left(\left(-2 \cdot J\right) \cdot \frac{\ell}{U} + \color{blue}{-1}\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot J, \frac{\ell}{U}, -1\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(U \cdot \mathsf{fma}\left(\color{blue}{J \cdot -2}, \frac{\ell}{U}, -1\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{neg}\left(U \cdot \mathsf{fma}\left(\color{blue}{J \cdot -2}, \frac{\ell}{U}, -1\right)\right) \]

       11. lower-/.f64 79.0

           \[\leadsto -U \cdot \mathsf{fma}\left(J \cdot -2, \color{blue}{\frac{\ell}{U}}, -1\right) \]

   11. Simplified 79.0%

       \[\leadsto \color{blue}{-U \cdot \mathsf{fma}\left(J \cdot -2, \frac{\ell}{U}, -1\right)} \]

   12. Taylor expanded in U around inf

       \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot U}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(U\right)\right)}\right) \]

       2. lower-neg.f64 70.8

          \[\leadsto -\color{blue}{\left(-U\right)} \]

   14. Simplified 70.8%

       \[\leadsto -\color{blue}{\left(-U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+38}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 74:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 72.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma J (* l (fma 0.3333333333333333 (* l l) 2.0)) U))

C

double code(double J, double l, double K, double U) {
	return fma(J, (l * fma(0.3333333333333333, (l * l), 2.0)), U);
}

Julia

function code(J, l, K, U)
	return fma(J, Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot {\ell}^{2}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   2. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   3. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{2}\right)} \cdot \cos \left(\frac{1}{2} \cdot K\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   4. *-commutative N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} \cdot J\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot K\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   5. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \left({\ell}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\ell \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

   8. +-commutative N/A

      \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)} + U \]

   9. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) + U \]

   10. associate-*r* N/A

       \[\leadsto \ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + {\ell}^{2} \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)}\right) + U \]

5. Simplified 81.7%

   \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)} + U \]

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. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right), U\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, U\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}, U\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}, U\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right), U\right) \]

   7. lower-*.f64 67.1

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, 2\right), U\right) \]

8. Simplified 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), U\right)} \]
9. Add Preprocessing

Alternative 22: 54.5% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \ell \cdot 2, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma J (* l 2.0) U))

C

double code(double J, double l, double K, double U) {
	return fma(J, (l * 2.0), U);
}

Julia

function code(J, l, K, U)
	return fma(J, Float64(l * 2.0), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   12. lower-*.f64 60.0

       \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

5. Simplified 60.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   4. *-commutative N/A

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   7. lower-*.f64 51.5

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

8. Simplified 51.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]
9. Add Preprocessing

Alternative 23: 37.0% 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 87.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(\left(2 \cdot J\right) \cdot \ell\right)} + U \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \left(2 \cdot J\right) \cdot \ell, U\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, \left(2 \cdot J\right) \cdot \ell, U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(J \cdot 2\right)} \cdot \ell, U\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{J \cdot \left(2 \cdot \ell\right)}, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   12. lower-*.f64 60.0

       \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

5. Simplified 60.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right) + U} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   4. *-commutative N/A

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} + U \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, 2 \cdot \ell, U\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

   7. lower-*.f64 51.5

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\ell \cdot 2}, U\right) \]

8. Simplified 51.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]

9. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]

    2. lower-neg.f64 N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{U \cdot \left(-2 \cdot \frac{J \cdot \ell}{U} - 1\right)}\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \color{blue}{\left(-2 \cdot \frac{J \cdot \ell}{U} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    5. associate-/l* N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \left(-2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \left(\color{blue}{\left(-2 \cdot J\right) \cdot \frac{\ell}{U}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \left(\left(-2 \cdot J\right) \cdot \frac{\ell}{U} + \color{blue}{-1}\right)\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot J, \frac{\ell}{U}, -1\right)}\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(U \cdot \mathsf{fma}\left(\color{blue}{J \cdot -2}, \frac{\ell}{U}, -1\right)\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \mathsf{neg}\left(U \cdot \mathsf{fma}\left(\color{blue}{J \cdot -2}, \frac{\ell}{U}, -1\right)\right) \]

    11. lower-/.f64 58.5

        \[\leadsto -U \cdot \mathsf{fma}\left(J \cdot -2, \color{blue}{\frac{\ell}{U}}, -1\right) \]

11. Simplified 58.5%

    \[\leadsto \color{blue}{-U \cdot \mathsf{fma}\left(J \cdot -2, \frac{\ell}{U}, -1\right)} \]

12. Taylor expanded in U around inf

    \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot U}\right) \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(U\right)\right)}\right) \]

    2. lower-neg.f64 36.4

       \[\leadsto -\color{blue}{\left(-U\right)} \]

14. Simplified 36.4%

    \[\leadsto -\color{blue}{\left(-U\right)} \]

15. Final simplification 36.4%

    \[\leadsto U \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))