Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 100.0%

Time: 10.7s

Alternatives: 18

Speedup: 1.5×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

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 87.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 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: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.96:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.96)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma
           l
           (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
           0.3333333333333333)
          2.0)))))
     (fma (* (sinh l) (* J 1.0)) 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 <= 0.96) {
		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((sinh(l) * (J * 1.0)), 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 <= 0.96)
		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(sinh(l) * Float64(J * 1.0)), 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, 0.96], 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[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.95999999999999996

   1. Initial program 87.4%

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

   3. Taylor expanded in l around 0

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

      1. 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 95.2

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

   5. Applied rewrites 95.2%

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

   if 0.95999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.96:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.96:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.96)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma (* l l) 0.016666666666666666 0.3333333333333333)
          2.0)))))
     (fma (* (sinh l) (* J 1.0)) 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 <= 0.96) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 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 <= 0.96)
		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(sinh(l) * Float64(J * 1.0)), 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, 0.96], 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[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.95999999999999996

   1. Initial program 87.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.95999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.96:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

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 (* (sinh l) (* J 1.0)) 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((sinh(l) * (J * 1.0)), 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(sinh(l) * Float64(J * 1.0)), 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[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $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 84.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 84.3

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

   5. Applied rewrites 84.3%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

4. Final simplification 94.6%

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

Alternative 5: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.04)
     (+ U (* t_0 (* l (* J (fma 0.3333333333333333 (* l l) 2.0)))))
     (fma (* (sinh l) (* J 1.0)) 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 * (l * (J * fma(0.3333333333333333, (l * l), 2.0))));
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 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(l * Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)))));
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 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[(l * N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $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 84.6%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

4. Final simplification 93.8%

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

Alternative 6: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

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 (* (sinh l) (* J 1.0)) 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((sinh(l) * (J * 1.0)), 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(sinh(l) * Float64(J * 1.0)), 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[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $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 84.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. Applied rewrites 80.9%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (cos (* K 0.5)) (* J (* l 2.0)) U)
   (fma (* (sinh l) (* J 1.0)) 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(cos((K * 0.5)), (J * (l * 2.0)), U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 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(cos(Float64(K * 0.5)), Float64(J * Float64(l * 2.0)), U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 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[(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[(J * 1.0), $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 84.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.0

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

   5. Applied rewrites 65.0%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    (*
     (* J (fma -0.125 (* K K) 1.0))
     (fma (* (* l l) 0.008333333333333333) (* l (* l l)) l))
    2.0
    U)
   (fma (* (sinh l) (* J 1.0)) 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(((J * fma(-0.125, (K * K), 1.0)) * fma(((l * l) * 0.008333333333333333), (l * (l * l)), l)), 2.0, U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 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(Float64(Float64(J * fma(-0.125, Float64(K * K), 1.0)) * fma(Float64(Float64(l * l) * 0.008333333333333333), Float64(l * Float64(l * l)), l)), 2.0, U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 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[(N[(N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)\right)}, 2, U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)}, 2, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) + 1\right)} \cdot \ell\right), 2, U\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell + \ell\right)}, 2, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}\right)} \cdot \ell + \ell\right), 2, U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + \ell\right), 2, U\right) \]

      6. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \color{blue}{{\ell}^{\left(2 + 1\right)}} + \ell\right), 2, U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{\color{blue}{3}} + \ell\right), 2, U\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)} + \ell\right), 2, U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right) + \ell\right), 2, U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}, \ell \cdot {\ell}^{2}, \ell\right)}, 2, U\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{\ell \cdot {\ell}^{2}}, \ell\right), 2, U\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \ell\right), 2, U\right) \]

      18. lower-*.f64 58.3

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \ell\right), 2, U\right) \]

   10. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}, 2, U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot {\ell}^{2}, \color{blue}{\ell} \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \color{blue}{\ell} \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right) \]

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    (*
     (* J (fma -0.125 (* K K) 1.0))
     (fma (* (* l l) 0.008333333333333333) (* l (* l l)) l))
    2.0
    U)
   (fma
    (*
     (* J 1.0)
     (fma
      (* l l)
      (* l (fma (* l l) (* (* l l) 0.0001984126984126984) 0.16666666666666666))
      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(((J * fma(-0.125, (K * K), 1.0)) * fma(((l * l) * 0.008333333333333333), (l * (l * l)), l)), 2.0, U);
	} else {
		tmp = fma(((J * 1.0) * fma((l * l), (l * fma((l * l), ((l * l) * 0.0001984126984126984), 0.16666666666666666)), 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(Float64(Float64(J * fma(-0.125, Float64(K * K), 1.0)) * fma(Float64(Float64(l * l) * 0.008333333333333333), Float64(l * Float64(l * l)), l)), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(l * l), Float64(l * fma(Float64(l * l), Float64(Float64(l * l) * 0.0001984126984126984), 0.16666666666666666)), 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[(N[(N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)\right)}, 2, U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)}, 2, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) + 1\right)} \cdot \ell\right), 2, U\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell + \ell\right)}, 2, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}\right)} \cdot \ell + \ell\right), 2, U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \left({\ell}^{2} \cdot \ell\right)} + \ell\right), 2, U\right) \]

      6. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \color{blue}{{\ell}^{\left(2 + 1\right)}} + \ell\right), 2, U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{\color{blue}{3}} + \ell\right), 2, U\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)} + \ell\right), 2, U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right) + \ell\right), 2, U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}, \ell \cdot {\ell}^{2}, \ell\right)}, 2, U\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot {\ell}^{2}, \ell\right), 2, U\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{\ell \cdot {\ell}^{2}}, \ell\right), 2, U\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{120}, \frac{1}{6}\right), \ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \ell\right), 2, U\right) \]

      18. lower-*.f64 58.3

          \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \ell\right), 2, U\right) \]

   10. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right)}, 2, U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot {\ell}^{2}, \color{blue}{\ell} \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \color{blue}{\ell} \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right) \]

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

   8. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 93.4

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

   10. Applied rewrites 93.4%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 93.4%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right) \cdot \ell, \ell\right), 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    (*
     (* J (fma -0.125 (* K K) 1.0))
     (fma l (* l (* l 0.16666666666666666)) l))
    2.0
    U)
   (fma
    (*
     (* J 1.0)
     (fma
      (* l l)
      (* l (fma (* l l) (* (* l l) 0.0001984126984126984) 0.16666666666666666))
      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(((J * fma(-0.125, (K * K), 1.0)) * fma(l, (l * (l * 0.16666666666666666)), l)), 2.0, U);
	} else {
		tmp = fma(((J * 1.0) * fma((l * l), (l * fma((l * l), ((l * l) * 0.0001984126984126984), 0.16666666666666666)), 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(Float64(Float64(J * fma(-0.125, Float64(K * K), 1.0)) * fma(l, Float64(l * Float64(l * 0.16666666666666666)), l)), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(l * l), Float64(l * fma(Float64(l * l), Float64(Float64(l * l) * 0.0001984126984126984), 0.16666666666666666)), 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[(N[(N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{6} \cdot {\ell}^{2}, \ell\right)}, 2, U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{{\ell}^{2} \cdot \frac{1}{6}}, \ell\right), 2, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{6}, \ell\right), 2, U\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{6}\right)}, \ell\right), 2, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{6}\right)}, \ell\right), 2, U\right) \]

      9. lower-*.f64 53.6

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)}, \ell\right), 2, U\right) \]

   10. Applied rewrites 53.6%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right)}, 2, U\right) \]

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

   8. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 93.4

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

   10. Applied rewrites 93.4%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 93.4%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right) \cdot \ell, \ell\right), 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma
    (*
     (* J (fma -0.125 (* K K) 1.0))
     (fma l (* l (* l 0.16666666666666666)) l))
    2.0
    U)
   (fma
    (* (* J 1.0) (fma (* (* l l) 0.008333333333333333) (* l (* l l)) 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(((J * fma(-0.125, (K * K), 1.0)) * fma(l, (l * (l * 0.16666666666666666)), l)), 2.0, U);
	} else {
		tmp = fma(((J * 1.0) * fma(((l * l) * 0.008333333333333333), (l * (l * l)), 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(Float64(Float64(J * fma(-0.125, Float64(K * K), 1.0)) * fma(l, Float64(l * Float64(l * 0.16666666666666666)), l)), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(Float64(l * l) * 0.008333333333333333), Float64(l * Float64(l * 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], -0.04], N[(N[(N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + 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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{6} \cdot {\ell}^{2}, \ell\right)}, 2, U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{{\ell}^{2} \cdot \frac{1}{6}}, \ell\right), 2, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{6}, \ell\right), 2, U\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{6}\right)}, \ell\right), 2, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(\frac{-1}{8}, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{6}\right)}, \ell\right), 2, U\right) \]

      9. lower-*.f64 53.6

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)}, \ell\right), 2, U\right) \]

   10. Applied rewrites 53.6%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right)}, 2, U\right) \]

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 91.9%

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

Alternative 12: 80.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.008333333333333333, \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (* J (fma -0.125 (* l (* K K)) l)) 2.0 U)
   (fma
    (* (* J 1.0) (fma (* (* l l) 0.008333333333333333) (* l (* l l)) 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((J * fma(-0.125, (l * (K * K)), l)), 2.0, U);
	} else {
		tmp = fma(((J * 1.0) * fma(((l * l) * 0.008333333333333333), (l * (l * l)), 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(Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l)), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(Float64(l * l) * 0.008333333333333333), Float64(l * Float64(l * 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], -0.04], N[(N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + 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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. Applied rewrites 28.6%

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

   8. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 51.9%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 91.9%

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

Alternative 13: 76.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\ell \cdot 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (* J (fma -0.125 (* l (* K K)) l)) 2.0 U)
   (fma (* (* J 1.0) (fma l (* l (* l 0.16666666666666666)) 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((J * fma(-0.125, (l * (K * K)), l)), 2.0, U);
	} else {
		tmp = fma(((J * 1.0) * fma(l, (l * (l * 0.16666666666666666)), 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(Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l)), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(l, Float64(l * Float64(l * 0.16666666666666666)), 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[(N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(l * N[(l * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. Applied rewrites 28.6%

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

   8. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 51.9%

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

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

   1. Initial program 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 97.7%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

Alternative 14: 59.2% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (* J (fma -0.125 (* l (* K K)) l)) 2.0 U)
   (fma J (* 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((J * fma(-0.125, (l * (K * K)), l)), 2.0, U);
	} else {
		tmp = fma(J, (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(Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l)), 2.0, U);
	else
		tmp = fma(J, Float64(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[(N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(l * 2.0), $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 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. Applied rewrites 28.6%

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

   8. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 51.9%

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

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

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 60.7%

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

Alternative 15: 58.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (* J l) (fma -0.25 (* K K) 2.0) U)
   (fma J (* 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((J * l), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(J, (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(Float64(J * l), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(J, Float64(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[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * 2.0), $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 84.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 50.2%

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

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

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 60.7%

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

Alternative 16: 55.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\left(J \cdot \ell\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (* (* J l) (fma -0.25 (* K K) 2.0))
   (fma J (* 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 = (J * l) * fma(-0.25, (K * K), 2.0);
	} else {
		tmp = fma(J, (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 = Float64(Float64(J * l) * fma(-0.25, Float64(K * K), 2.0));
	else
		tmp = fma(J, Float64(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[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * 2.0), $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 84.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in J around inf

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

   8. Applied rewrites 37.6%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 48.6%

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

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

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 60.7%

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

Alternative 17: 54.0% 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 87.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 63.4

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 53.5%

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

Alternative 18: 19.6% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 63.4

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 53.5%

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

9. Taylor expanded in J around inf

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

11. Applied rewrites 19.2%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))