Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 97.3%

Time: 14.7s

Alternatives: 19

Speedup: 2.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 (- INFINITY))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          l
          (*
           l
           (fma
            (* l l)
            (fma (* l l) 0.0003968253968253968 0.016666666666666666)
            0.3333333333333333))
          2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * fma(l, (l * fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(l, Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 78.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 98.5%

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

Alternative 2: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.405:\\ \;\;\;\;J \cdot \mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot 2, \frac{U}{J}\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right) \cdot \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(0.0003968253968253968 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right)\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.405)
     (* J (fma (cos (* K 0.5)) (* l 2.0) (/ U J)))
     (if (<= t_0 -0.01)
       (fma
        l
        (*
         (fma
          (* l l)
          (fma l (* l 0.016666666666666666) 0.3333333333333333)
          2.0)
         (fma (* K K) (* J -0.125) J))
        U)
       (fma
        (* l (fma l (* l (* 0.0003968253968253968 (* (* l l) (* l l)))) 2.0))
        J
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.405) {
		tmp = J * fma(cos((K * 0.5)), (l * 2.0), (U / J));
	} else if (t_0 <= -0.01) {
		tmp = fma(l, (fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0) * fma((K * K), (J * -0.125), J)), U);
	} else {
		tmp = fma((l * fma(l, (l * (0.0003968253968253968 * ((l * l) * (l * l)))), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.405)
		tmp = Float64(J * fma(cos(Float64(K * 0.5)), Float64(l * 2.0), Float64(U / J)));
	elseif (t_0 <= -0.01)
		tmp = fma(l, Float64(fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0) * fma(Float64(K * K), Float64(J * -0.125), J)), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * Float64(0.0003968253968253968 * Float64(Float64(l * l) * Float64(l * l)))), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.405], N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(l * N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(J * -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.40500000000000003

   1. Initial program 82.5%

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.6

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

   8. Simplified 67.6%

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

   if -0.40500000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 95.6%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.5

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

   8. Simplified 73.5%

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

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

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.4

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 91.9

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

   11. Simplified 91.9%

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

   12. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{2520} \cdot {\ell}^{5}}, 2\right), J, U\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot {\ell}^{\color{blue}{\left(4 + 1\right)}}, 2\right), J, U\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot \color{blue}{\left({\ell}^{4} \cdot \ell\right)}, 2\right), J, U\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right) \cdot \ell}, 2\right), J, U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 91.9

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

   14. Simplified 91.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.405:\\ \;\;\;\;J \cdot \mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot 2, \frac{U}{J}\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right) \cdot \mathsf{fma}\left(K \cdot K, J \cdot -0.125, J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(0.0003968253968253968 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right)\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l))));
	else
		tmp = fma(Float64(l * fma(l, Float64(l * Float64(0.0003968253968253968 * Float64(Float64(l * l) * Float64(l * l)))), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \ell\right) \cdot {K}^{2}\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right)} \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

   8. Simplified 61.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)} + U \]

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

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.4

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 91.9

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

   11. Simplified 91.9%

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

   12. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{2520} \cdot {\ell}^{5}}, 2\right), J, U\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot {\ell}^{\color{blue}{\left(4 + 1\right)}}, 2\right), J, U\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot \color{blue}{\left({\ell}^{4} \cdot \ell\right)}, 2\right), J, U\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right) \cdot \ell}, 2\right), J, U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 91.9

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

   14. Simplified 91.9%

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

4. Final simplification 83.6%

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

Alternative 4: 83.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(l, Float64(fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0) * fma(Float64(K * K), Float64(J * -0.125), J)), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * Float64(0.0003968253968253968 * Float64(Float64(l * l) * Float64(l * l)))), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(l * N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(J * -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 60.1

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

   8. Simplified 60.1%

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

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

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.4

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 91.9

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

   11. Simplified 91.9%

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

   12. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{2520} \cdot {\ell}^{5}}, 2\right), J, U\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot {\ell}^{\color{blue}{\left(4 + 1\right)}}, 2\right), J, U\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot \color{blue}{\left({\ell}^{4} \cdot \ell\right)}, 2\right), J, U\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right) \cdot \ell}, 2\right), J, U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 91.9

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

   14. Simplified 91.9%

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

4. Final simplification 83.2%

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

Alternative 5: 95.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (*
    J
    (*
     l
     (fma
      l
      (*
       l
       (fma
        (* l l)
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        0.3333333333333333))
      2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * fma(l, (l * fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0))));
}

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(l, Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 95.9

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

5. Simplified 95.9%

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

6. Final simplification 95.9%

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

Alternative 6: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 2e+14)
   (+
    U
    (*
     (fma
      l
      (*
       l
       (fma
        l
        (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
        0.3333333333333333))
      2.0)
     (* J (fma -0.125 (* l (* K K)) l))))
   (fma
    l
    (*
     (fma (* l l) (fma l (* l 0.016666666666666666) 0.3333333333333333) 2.0)
     (* J (cos (* K 0.5))))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 2e+14) {
		tmp = U + (fma(l, (l * fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * (J * fma(-0.125, (l * (K * K)), l)));
	} else {
		tmp = fma(l, (fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0) * (J * cos((K * 0.5)))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 2e+14)
		tmp = Float64(U + Float64(fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l))));
	else
		tmp = fma(l, Float64(fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0) * Float64(J * cos(Float64(K * 0.5)))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 2e+14], N[(U + N[(N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2e14

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \ell\right) \cdot {K}^{2}\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right)} \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)} + U \]

   if 2e14 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.114:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.5, \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(0.0003968253968253968 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right)\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.114)
   (fma (fma l (* l 0.5) l) J U)
   (fma
    (* l (fma l (* l (* 0.0003968253968253968 (* (* l l) (* l l)))) 2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.114) {
		tmp = fma(fma(l, (l * 0.5), l), J, U);
	} else {
		tmp = fma((l * fma(l, (l * (0.0003968253968253968 * ((l * l) * (l * l)))), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.114)
		tmp = fma(fma(l, Float64(l * 0.5), l), J, U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * Float64(0.0003968253968253968 * Float64(Float64(l * l) * Float64(l * l)))), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.114], N[(N[(l * N[(l * 0.5), $MachinePrecision] + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.114000000000000004

   1. Initial program 83.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 30.6

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 31.3%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 60.0

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

   11. Simplified 60.0%

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

   if 0.114000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.7

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 92.2%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 92.2

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

   11. Simplified 92.2%

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

   12. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{2520} \cdot {\ell}^{5}}, 2\right), J, U\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot {\ell}^{\color{blue}{\left(4 + 1\right)}}, 2\right), J, U\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \frac{1}{2520} \cdot \color{blue}{\left({\ell}^{4} \cdot \ell\right)}, 2\right), J, U\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right) \cdot \ell}, 2\right), J, U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{4}\right)}, 2\right), J, U\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 92.2

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

   14. Simplified 92.2%

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

Alternative 8: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 2e+14)
   (+
    U
    (*
     (fma
      l
      (*
       l
       (fma
        l
        (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
        0.3333333333333333))
      2.0)
     (* J (fma -0.125 (* l (* K K)) l))))
   (+ U (* (cos (/ K 2.0)) (* J (* l (fma l (* l 0.3333333333333333) 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 2e+14) {
		tmp = U + (fma(l, (l * fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * (J * fma(-0.125, (l * (K * K)), l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * fma(l, (l * 0.3333333333333333), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 2e+14)
		tmp = Float64(U + Float64(fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 2e+14], N[(U + N[(N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2e14

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \ell\right) \cdot {K}^{2}\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right)} \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)} + U \]

   if 2e14 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (*
    J
    (*
     l
     (fma
      (* l l)
      (fma l (* l 0.016666666666666666) 0.3333333333333333)
      2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0))));
}

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 94.0

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

5. Simplified 94.0%

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

6. Final simplification 94.0%

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

Alternative 10: 78.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.114:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.5, \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.114)
   (fma (fma l (* l 0.5) l) J U)
   (fma
    (*
     l
     (fma l (* l (fma l (* l 0.016666666666666666) 0.3333333333333333)) 2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.114) {
		tmp = fma(fma(l, (l * 0.5), l), J, U);
	} else {
		tmp = fma((l * fma(l, (l * fma(l, (l * 0.016666666666666666), 0.3333333333333333)), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.114)
		tmp = fma(fma(l, Float64(l * 0.5), l), J, U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.114], N[(N[(l * N[(l * 0.5), $MachinePrecision] + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.114000000000000004

   1. Initial program 83.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 30.6

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 31.3%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 60.0

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

   11. Simplified 60.0%

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

   if 0.114000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.7

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 92.2%

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

   9. Taylor expanded in l around 0

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

   11. Simplified 90.2%

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

Alternative 11: 77.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+28}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e+28)
   (+
    U
    (*
     (fma
      l
      (*
       l
       (fma
        l
        (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
        0.3333333333333333))
      2.0)
     (* J (fma -0.125 (* l (* K K)) l))))
   (fma (* (cos (* K 0.5)) (* l J)) (fma l (* l 0.3333333333333333) 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e+28) {
		tmp = U + (fma(l, (l * fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * (J * fma(-0.125, (l * (K * K)), l)));
	} else {
		tmp = fma((cos((K * 0.5)) * (l * J)), fma(l, (l * 0.3333333333333333), 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e+28)
		tmp = Float64(U + Float64(fma(l, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333)), 2.0) * Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l))));
	else
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * J)), fma(l, Float64(l * 0.3333333333333333), 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e+28], N[(U + N[(N[(l * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 9.99999999999999958e27

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)} + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \ell\right) \cdot {K}^{2}\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right)} \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

   8. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)} + U \]

   if 9.99999999999999958e27 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 82.0%

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

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

   5. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right), \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+28}:\\ \;\;\;\;U + \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right), \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{elif}\;\ell \leq -1.9:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{elif}\;\ell \leq 85:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \mathsf{expm1}\left(\ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2e+209)
   (fma
    (* J (fma -0.125 (* l (* K K)) l))
    (fma l (* l 0.3333333333333333) 2.0)
    U)
   (if (<= l -1.9)
     (fma (- 1.0 (exp (- l))) J U)
     (if (<= l 85.0)
       (fma l (* (cos (* K 0.5)) (* J 2.0)) U)
       (* J (expm1 l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2e+209) {
		tmp = fma((J * fma(-0.125, (l * (K * K)), l)), fma(l, (l * 0.3333333333333333), 2.0), U);
	} else if (l <= -1.9) {
		tmp = fma((1.0 - exp(-l)), J, U);
	} else if (l <= 85.0) {
		tmp = fma(l, (cos((K * 0.5)) * (J * 2.0)), U);
	} else {
		tmp = J * expm1(l);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2e+209)
		tmp = fma(Float64(J * fma(-0.125, Float64(l * Float64(K * K)), l)), fma(l, Float64(l * 0.3333333333333333), 2.0), U);
	elseif (l <= -1.9)
		tmp = fma(Float64(1.0 - exp(Float64(-l))), J, U);
	elseif (l <= 85.0)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0)), U);
	else
		tmp = Float64(J * expm1(l));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2e+209], N[(N[(J * N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, -1.9], N[(N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 85.0], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.0000000000000001e209

   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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.8

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

   8. Simplified 78.8%

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

   if -2.0000000000000001e209 < l < -1.8999999999999999

   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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 75.5

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1} - e^{\mathsf{neg}\left(\ell\right)}, J, U\right) \]
   7. Step-by-step derivation

   8. Simplified 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1} - e^{-\ell}, J, U\right) \]

   if -1.8999999999999999 < l < 85

   1. Initial program 68.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   if 85 < 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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 73.6

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 73.6%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in J around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - 1\right)} \]

       2. lower-expm1.f64 73.6

          \[\leadsto J \cdot \color{blue}{\mathsf{expm1}\left(\ell\right)} \]

   11. Simplified 73.6%

       \[\leadsto \color{blue}{J \cdot \mathsf{expm1}\left(\ell\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \mathsf{fma}\left(-0.125, \ell \cdot \left(K \cdot K\right), \ell\right), \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{elif}\;\ell \leq -1.9:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{elif}\;\ell \leq 85:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \mathsf{expm1}\left(\ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.114:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.5, \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.114)
   (fma (fma l (* l 0.5) l) J U)
   (fma (* l (fma l (* l 0.3333333333333333) 2.0)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.114) {
		tmp = fma(fma(l, (l * 0.5), l), J, U);
	} else {
		tmp = fma((l * fma(l, (l * 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.114)
		tmp = fma(fma(l, Float64(l * 0.5), l), J, U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.114], N[(N[(l * N[(l * 0.5), $MachinePrecision] + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.114000000000000004

   1. Initial program 83.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 30.6

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 31.3%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 60.0

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

   11. Simplified 60.0%

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

   if 0.114000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.7

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 92.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 86.0

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

   11. Simplified 86.0%

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

Alternative 14: 73.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.114:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.5, \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.114)
   (fma (fma l (* l 0.5) l) 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)) <= 0.114) {
		tmp = fma(fma(l, (l * 0.5), l), 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)) <= 0.114)
		tmp = fma(fma(l, Float64(l * 0.5), l), J, U);
	else
		tmp = fma(l, Float64(J * fma(Float64(l * 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], 0.114], N[(N[(l * N[(l * 0.5), $MachinePrecision] + l), $MachinePrecision] * J + U), $MachinePrecision], N[(l * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.114000000000000004

   1. Initial program 83.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 30.6

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 31.3%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 60.0

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

   11. Simplified 60.0%

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

   if 0.114000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

   5. Simplified 85.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Simplified 81.4%

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

Alternative 15: 58.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.478:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.5, \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.478)
   (fma (fma l (* l 0.5) l) J U)
   (fma (* l 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.478) {
		tmp = fma(fma(l, (l * 0.5), l), J, U);
	} else {
		tmp = fma((l * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.478)
		tmp = fma(fma(l, Float64(l * 0.5), l), J, U);
	else
		tmp = fma(Float64(l * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.478], N[(N[(l * N[(l * 0.5), $MachinePrecision] + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.47799999999999998

   1. Initial program 84.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 41.3

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 39.8%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 61.1

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

   11. Simplified 61.1%

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

   if 0.47799999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.7

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.1

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

   8. Simplified 61.1%

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

Alternative 16: 57.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.478:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot J, 0.5, J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.478)
   (fma l (fma (* l J) 0.5 J) U)
   (fma (* l 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.478) {
		tmp = fma(l, fma((l * J), 0.5, J), U);
	} else {
		tmp = fma((l * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.478)
		tmp = fma(l, fma(Float64(l * J), 0.5, J), U);
	else
		tmp = fma(Float64(l * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.478], N[(l * N[(N[(l * J), $MachinePrecision] * 0.5 + J), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.47799999999999998

   1. Initial program 84.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 41.3

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

   8. Simplified 39.8%

      \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 50.7

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

   11. Simplified 50.7%

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

   if 0.47799999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 85.7

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.1

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

   8. Simplified 61.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.478:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot J, 0.5, J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.5% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 70.2

      \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

5. Simplified 70.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

6. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.8

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

8. Simplified 51.8%

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

Alternative 18: 54.5% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 70.2

      \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

5. Simplified 70.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

6. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

8. Simplified 74.9%

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

9. Taylor expanded in l around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 51.8

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

11. Simplified 51.8%

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

12. Final simplification 51.8%

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

Alternative 19: 48.5% accurate, 47.1× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (fma J l U))

C

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

Julia

function code(J, l, K, U)
	return fma(J, l, U)
end

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 70.2

      \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\color{blue}{-\ell}}, J, U\right) \]

5. Simplified 70.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\ell} - e^{-\ell}, J, U\right)} \]

6. Taylor expanded in l around 0

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

8. Simplified 48.7%

   \[\leadsto \mathsf{fma}\left(e^{\ell} - \color{blue}{1}, J, U\right) \]

9. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + J \cdot \ell} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{J \cdot \ell + U} \]

    2. lower-fma.f64 46.0

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

11. Simplified 46.0%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))