Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.9%

Time: 13.2s

Alternatives: 22

Speedup: 1.5×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. lift-+.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: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.17)
     (+ U (* t_0 (* J (* l (fma l (* l 0.3333333333333333) 2.0)))))
     (if (<= t_0 -0.01)
       (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
       (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 <= -0.17) {
		tmp = U + (t_0 * (J * (l * fma(l, (l * 0.3333333333333333), 2.0))));
	} else if (t_0 <= -0.01) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} 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 <= -0.17)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)))));
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], N[(U + N[(t$95$0 * N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.170000000000000012

   1. Initial program 88.4%

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

   3. Taylor expanded in l around 0

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

      1. 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -0.170000000000000012 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 100.0%

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

      1. lift-+.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

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot \sinh \ell, 2, U\right) \]
   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 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.17:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.17)
     (fma (* (cos (* K 0.5)) (* J (fma l (* l 0.3333333333333333) 2.0))) l U)
     (if (<= t_0 -0.01)
       (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
       (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 <= -0.17) {
		tmp = fma((cos((K * 0.5)) * (J * fma(l, (l * 0.3333333333333333), 2.0))), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} 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 <= -0.17)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.170000000000000012

   1. Initial program 88.4%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.9%

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

   if -0.170000000000000012 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 100.0%

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

      1. lift-+.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

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot \sinh \ell, 2, U\right) \]
   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 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.17)
     (fma l (* (* J (cos (* K 0.5))) (fma l (* l 0.3333333333333333) 2.0)) U)
     (if (<= t_0 -0.01)
       (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
       (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 <= -0.17) {
		tmp = fma(l, ((J * cos((K * 0.5))) * fma(l, (l * 0.3333333333333333), 2.0)), U);
	} else if (t_0 <= -0.01) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} 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 <= -0.17)
		tmp = fma(l, Float64(Float64(J * cos(Float64(K * 0.5))) * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], N[(l * N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.170000000000000012

   1. Initial program 88.4%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -0.170000000000000012 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 100.0%

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

      1. lift-+.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

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot \sinh \ell, 2, U\right) \]
   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 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (* (cos (* K 0.5)) (* J 2.0)) l U)
     (if (<= t_0 -0.01)
       (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
       (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 <= -0.42) {
		tmp = fma((cos((K * 0.5)) * (J * 2.0)), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} 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 <= -0.42)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0)), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.6%

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

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. 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

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot \sinh \ell, 2, U\right) \]
   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 82.5

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

   7. Applied rewrites 82.5%

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

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right) \cdot \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (* (cos (* K 0.5)) (* J 2.0)) l U)
     (if (<= t_0 -0.01)
       (fma
        (*
         (fma
          (* l (* l l))
          (fma
           (* l l)
           (fma (* l l) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666)
          l)
         (fma -0.125 (* J (* K K)) J))
        2.0
        U)
       (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 <= -0.42) {
		tmp = fma((cos((K * 0.5)) * (J * 2.0)), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma((fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, (J * (K * K)), J)), 2.0, U);
	} 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 <= -0.42)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0)), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, Float64(J * Float64(K * K)), J)), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.6%

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

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. 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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + \frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right) + J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, J \cdot {K}^{2}, J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{J \cdot {K}^{2}}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      5. lower-*.f64 73.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

   10. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right) \cdot \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right) \cdot \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (cos (* K 0.5)) (* J (* l 2.0)) U)
     (if (<= t_0 -0.01)
       (fma
        (*
         (fma
          (* l (* l l))
          (fma
           (* l l)
           (fma (* l l) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666)
          l)
         (fma -0.125 (* J (* K K)) J))
        2.0
        U)
       (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 <= -0.42) {
		tmp = fma(cos((K * 0.5)), (J * (l * 2.0)), U);
	} else if (t_0 <= -0.01) {
		tmp = fma((fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, (J * (K * K)), J)), 2.0, U);
	} 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 <= -0.42)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(l * 2.0)), U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, Float64(J * Float64(K * K)), J)), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   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 77.6

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

   5. Applied rewrites 77.6%

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

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. 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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + \frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right) + J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, J \cdot {K}^{2}, J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{J \cdot {K}^{2}}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      5. lower-*.f64 73.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

   10. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right) \cdot \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.98:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.98)
     (+
      U
      (*
       t_0
       (*
        l
        (*
         J
         (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 <= 0.98) {
		tmp = U + (t_0 * (l * (J * 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 <= 0.98)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 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, 0.98], N[(U + N[(t$95$0 * N[(l * N[(J * 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))) < 0.97999999999999998

   1. Initial program 90.4%

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

   3. Taylor expanded in l around 0

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

      1. 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 91.3%

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

   if 0.97999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.5%

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

      1. lift-+.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 99.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), J \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.98)
   (fma
    (cos (* K 0.5))
    (*
     l
     (fma
      (* l l)
      (* J (fma (* l l) 0.016666666666666666 0.3333333333333333))
      (* J 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)) <= 0.98) {
		tmp = fma(cos((K * 0.5)), (l * fma((l * l), (J * fma((l * l), 0.016666666666666666, 0.3333333333333333)), (J * 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)) <= 0.98)
		tmp = fma(cos(Float64(K * 0.5)), Float64(l * fma(Float64(l * l), Float64(J * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), Float64(J * 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], 0.98], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $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))) < 0.97999999999999998

   1. Initial program 90.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if 0.97999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.5%

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

      1. lift-+.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 99.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), J \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     (fma
      (* l (* l l))
      (fma
       (* l l)
       (fma (* l l) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      l)
     (fma -0.125 (* J (* K K)) J))
    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)) <= -0.01) {
		tmp = fma((fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, (J * (K * K)), J)), 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)) <= -0.01)
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l) * fma(-0.125, Float64(J * Float64(K * K)), J)), 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], -0.01], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $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))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. 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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + \frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right) + J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, J \cdot {K}^{2}, J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{J \cdot {K}^{2}}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      5. lower-*.f64 62.2

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.0%

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

4. Final simplification 88.3%

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

Alternative 11: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. lift-+.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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 12: 82.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (fma
          (* l (* l l))
          (fma
           (* l l)
           (fma (* l l) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666)
          l)))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma (* t_0 (fma -0.125 (* J (* K K)) J)) 2.0 U)
     (fma (* t_0 (* J 1.0)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l);
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((t_0 * fma(-0.125, (J * (K * K)), J)), 2.0, U);
	} else {
		tmp = fma((t_0 * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(t_0 * fma(-0.125, Float64(J * Float64(K * K)), J)), 2.0, U);
	else
		tmp = fma(Float64(t_0 * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(t$95$0 * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(t$95$0 * 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))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. 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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + \frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right) + J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, J \cdot {K}^{2}, J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{J \cdot {K}^{2}}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \ell\right), 2, U\right) \]

      5. lower-*.f64 62.2

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, J \cdot \color{blue}{\left(K \cdot K\right)}, J\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right)} \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{1}\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

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

Alternative 13: 82.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 20000.0)
   (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
   (fma
    (*
     (* J (cos (* K 0.5)))
     (fma
      (* l (* l l))
      (fma
       (* l l)
       (fma (* l l) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      l))
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 20000.0) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} else {
		tmp = fma(((J * cos((K * 0.5))) * fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 20000.0)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 2.0, U);
	else
		tmp = fma(Float64(Float64(J * cos(Float64(K * 0.5))) * fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 20000.0], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2e4

   1. Initial program 90.7%

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

      1. 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

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot \sinh \ell, 2, U\right) \]
   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 85.9

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

   7. Applied rewrites 85.9%

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

   if 2e4 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 92.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right)}, 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (* l (fma l (* l 0.3333333333333333) 2.0))
    (fma -0.125 (* J (* K K)) J)
    U)
   (fma
    (*
     (fma
      (* l (* l l))
      (fma
       (* l l)
       (fma (* l l) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      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)) <= -0.01) {
		tmp = fma((l * fma(l, (l * 0.3333333333333333), 2.0)), fma(-0.125, (J * (K * K)), J), U);
	} else {
		tmp = fma((fma((l * (l * l)), fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 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)) <= -0.01)
		tmp = fma(Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), fma(-0.125, Float64(J * Float64(K * K)), J), U);
	else
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 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], -0.01], N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 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))) < -0.0100000000000000002

   1. Initial program 90.4%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + \frac{1}{384} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]

   8. Applied rewrites 52.4%

      \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\left(K \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right)\right)\right)\right)}, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right)\right), U\right) \]

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 59.4%

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

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

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(\left(J \cdot \color{blue}{1}\right) \cdot \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell\right), 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

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

Alternative 15: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot t\_0, \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot t\_0, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma l (* l 0.3333333333333333) 2.0)))
   (if (<= (cos (/ K 2.0)) 0.055)
     (fma (* l t_0) (fma -0.125 (* J (* K K)) J) U)
     (fma l (* J t_0) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma(l, (l * 0.3333333333333333), 2.0);
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma((l * t_0), fma(-0.125, (J * (K * K)), J), U);
	} else {
		tmp = fma(l, (J * t_0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(l, Float64(l * 0.3333333333333333), 2.0)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(l * t_0), fma(-0.125, Float64(J * Float64(K * K)), J), U);
	else
		tmp = fma(l, Float64(J * t_0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(l * t$95$0), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * t$95$0), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + \frac{1}{384} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\left(K \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right)\right)\right)\right)}, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right)\right), U\right) \]

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 58.5%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 86.1%

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

Alternative 16: 73.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma (* l (fma (* K K) -0.25 2.0)) J U)
   (fma l (* J (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	} else {
		tmp = fma(l, (J * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 54.5%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 86.1%

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

Alternative 17: 72.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma (* J l) (fma -0.25 (* K K) 2.0) U)
   (fma l (* J (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma((J * l), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(l, (J * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(J * l), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 51.7%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 86.1%

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

Alternative 18: 72.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma (* J l) (fma -0.25 (* K K) 2.0) U)
   (fma (* J l) (fma l (* l 0.3333333333333333) 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma((J * l), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma((J * l), fma(l, (l * 0.3333333333333333), 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(J * l), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(Float64(J * l), fma(l, Float64(l * 0.3333333333333333), 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * l), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 51.7%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 86.1%

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

Alternative 19: 71.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\ell, t\_0 \cdot \left(J \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(U, 2 \cdot \frac{J \cdot \ell}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot t\_0, \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma l (* l 0.3333333333333333) 2.0)))
   (if (<= l -9.8e-8)
     (fma
      l
      (*
       t_0
       (* J (fma K (* K (fma (* K K) 0.0026041666666666665 -0.125)) 1.0)))
      U)
     (if (<= l 1.8e+51)
       (fma U (* 2.0 (/ (* J l) U)) U)
       (fma (* l t_0) (fma -0.125 (* J (* K K)) J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma(l, (l * 0.3333333333333333), 2.0);
	double tmp;
	if (l <= -9.8e-8) {
		tmp = fma(l, (t_0 * (J * fma(K, (K * fma((K * K), 0.0026041666666666665, -0.125)), 1.0))), U);
	} else if (l <= 1.8e+51) {
		tmp = fma(U, (2.0 * ((J * l) / U)), U);
	} else {
		tmp = fma((l * t_0), fma(-0.125, (J * (K * K)), J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(l, Float64(l * 0.3333333333333333), 2.0)
	tmp = 0.0
	if (l <= -9.8e-8)
		tmp = fma(l, Float64(t_0 * Float64(J * fma(K, Float64(K * fma(Float64(K * K), 0.0026041666666666665, -0.125)), 1.0))), U);
	elseif (l <= 1.8e+51)
		tmp = fma(U, Float64(2.0 * Float64(Float64(J * l) / U)), U);
	else
		tmp = fma(Float64(l * t_0), fma(-0.125, Float64(J * Float64(K * K)), J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[l, -9.8e-8], N[(l * N[(t$95$0 * N[(J * N[(K * N[(K * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.8e+51], N[(U * N[(2.0 * N[(N[(J * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * t$95$0), $MachinePrecision] * N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.8000000000000004e-8

   1. Initial program 99.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right), U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right) \]

   if -9.8000000000000004e-8 < l < 1.80000000000000005e51

   1. Initial program 83.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 84.6%

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

   if 1.80000000000000005e51 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + \frac{1}{384} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]

   8. Applied rewrites 50.3%

      \[\leadsto \mathsf{fma}\left(\ell, \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\left(K \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665\right)\right)\right)\right)}, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right)\right), U\right) \]

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 77.1%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \color{blue}{\mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right)}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(J \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(U, 2 \cdot \frac{J \cdot \ell}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), \mathsf{fma}\left(-0.125, J \cdot \left(K \cdot K\right), J\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 69.1% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 85.8%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 73.2%

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

Alternative 21: 53.4% 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 91.1%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 67.1

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

5. Applied rewrites 67.1%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 58.4%

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

Alternative 22: 19.2% 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 91.1%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 67.1

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

5. Applied rewrites 67.1%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 58.4%

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

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 18.3%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))