Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 100.0%

Time: 13.3s

Alternatives: 19

Speedup: 1.8×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sinh-undef N/A

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

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

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

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

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

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

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

   9. div-inv N/A

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

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

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

   11. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.9:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.9)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma (* l l) 0.016666666666666666 0.3333333333333333)
          2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.9) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.9)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.9], N[(U + N[(t$95$0 * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.900000000000000022

   1. Initial program 86.8%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 95.2

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

   5. Simplified 95.2%

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

   if 0.900000000000000022 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 98.5%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 98.5

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

   9. Applied egg-rr 98.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.9:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.866:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.866)
     (+ U (* t_0 (* J (* l (fma 0.3333333333333333 (* l l) 2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.866) {
		tmp = U + (t_0 * (J * (l * fma(0.3333333333333333, (l * l), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.866)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.866], N[(U + N[(t$95$0 * N[(J * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.865999999999999992

   1. Initial program 87.1%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 94.9

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

   5. Simplified 94.9%

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

   if 0.865999999999999992 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 98.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 98.0

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

   9. Applied egg-rr 98.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.866:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.866:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.866)
   (fma l (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (* l l) 2.0))) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.866) {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(0.3333333333333333, (l * l), 2.0))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.866)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.866], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.865999999999999992

   1. Initial program 87.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 90.7%

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

   if 0.865999999999999992 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 98.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 98.0

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

   9. Applied egg-rr 98.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.866:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= (cos (/ K 2.0)) -0.02)
     (fma (* t_0 (fma -0.125 (* K K) 1.0)) J U)
     (fma t_0 J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((t_0 * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(t_0 * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(t$95$0 * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.8

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

   7. Simplified 71.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 94.8

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

   9. Applied egg-rr 94.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -\infty:\\ \;\;\;\;J \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* J (- (exp l) (exp (- l)))) (- INFINITY))
   (* J (* l (fma 0.3333333333333333 (* l l) 2.0)))
   (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 ((J * (exp(l) - exp(-l))) <= -((double) INFINITY)) {
		tmp = J * (l * fma(0.3333333333333333, (l * l), 2.0));
	} 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 (Float64(J * Float64(exp(l) - exp(Float64(-l)))) <= Float64(-Inf))
		tmp = Float64(J * Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0)));
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 72.4%

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

   6. Taylor expanded in K around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   9. Taylor expanded in J around inf

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

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

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

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

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

       3. +-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. /-lowering-/.f64 54.2

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

   11. Simplified 54.2%

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

   12. Taylor expanded in l around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

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

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

       10. +-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 52.9

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

   14. Simplified 52.9%

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

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

   1. Initial program 85.9%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 93.2%

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

   6. Taylor expanded in K around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.7

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

   8. Simplified 76.7%

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

Alternative 7: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, K \cdot \left(K \cdot \left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    l
    (*
     K
     (*
      K
      (* (fma 0.3333333333333333 (* l l) 2.0) (fma J -0.125 (/ J (* K K))))))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(l, (K * (K * (fma(0.3333333333333333, (l * l), 2.0) * fma(J, -0.125, (J / (K * K)))))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(l, Float64(K * Float64(K * Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * fma(J, -0.125, Float64(J / Float64(K * K)))))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(l * N[(K * N[(K * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * -0.125 + N[(J / N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.8

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

   8. Simplified 65.8%

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

   9. Taylor expanded in K around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. distribute-lft-out N/A

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

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

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

   11. Simplified 69.0%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 94.8

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

   9. Applied egg-rr 94.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, K \cdot \left(K \cdot \left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-71)
   (fma (* 2.0 (sinh l)) J U)
   (+
    U
    (*
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma
         (* l l)
         (fma (* l l) 0.0003968253968253968 0.016666666666666666)
         0.3333333333333333)
        2.0)))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-71) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = U + ((J * (l * fma((l * l), fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0))) * cos((K / 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-71)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-71], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 9.9999999999999992e-72

   1. Initial program 90.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 83.3%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. sinh-lowering-sinh.f64 83.3

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

   9. Applied egg-rr 83.3%

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

   if 9.9999999999999992e-72 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 97.5

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

   5. Simplified 97.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, K \cdot \left(K \cdot \left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \mathsf{fma}\left(J, -0.125, \frac{J}{K \cdot K}\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    l
    (*
     K
     (*
      K
      (* (fma 0.3333333333333333 (* l l) 2.0) (fma J -0.125 (/ J (* K K))))))
    U)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       (* l l)
       (fma l (* l 0.0003968253968253968) 0.016666666666666666)
       0.3333333333333333)
      2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(l, (K * (K * (fma(0.3333333333333333, (l * l), 2.0) * fma(J, -0.125, (J / (K * K)))))), U);
	} else {
		tmp = fma((l * fma((l * l), fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(l, Float64(K * Float64(K * Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * fma(J, -0.125, Float64(J / Float64(K * K)))))), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(l * N[(K * N[(K * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * N[(J * -0.125 + N[(J / N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 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))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.8

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

   8. Simplified 65.8%

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

   9. Taylor expanded in K around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. distribute-lft-out N/A

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

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

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

   11. Simplified 69.0%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 89.8

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

   10. Simplified 89.8%

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

Alternative 10: 83.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    l
    (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma -0.125 (* K K) 1.0))
    U)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       (* l l)
       (fma l (* l 0.0003968253968253968) 0.016666666666666666)
       0.3333333333333333)
      2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(l, ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(-0.125, (K * K), 1.0)), U);
	} else {
		tmp = fma((l * fma((l * l), fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(l, Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(-0.125, Float64(K * K), 1.0)), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 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))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.8

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

   8. Simplified 65.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 89.8

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

   10. Simplified 89.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    l
    (* (* J (fma 0.3333333333333333 (* l l) 2.0)) (fma -0.125 (* K K) 1.0))
    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)) <= -0.02) {
		tmp = fma(l, ((J * fma(0.3333333333333333, (l * l), 2.0)) * fma(-0.125, (K * K), 1.0)), 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)) <= -0.02)
		tmp = fma(l, Float64(Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)) * fma(-0.125, Float64(K * K), 1.0)), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(l * N[(N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 65.8

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

   8. Simplified 65.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 88.3

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

   10. Simplified 88.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l J) (fma -0.25 (* K K) 2.0) 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)) <= -0.02) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), 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)) <= -0.02)
		tmp = fma(Float64(l * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 57.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

       10. *-commutative N/A

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 60.3

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

   11. Simplified 60.3%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 88.3

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

   10. Simplified 88.3%

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

Alternative 13: 76.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l J) (fma -0.25 (* K K) 2.0) U)
   (fma (* l (fma 0.3333333333333333 (* l l) 2.0)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma((l * fma(0.3333333333333333, (l * l), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(l * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(Float64(l * fma(0.3333333333333333, 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], -0.02], N[(N[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(0.3333333333333333 * N[(l * l), $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))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 57.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

       10. *-commutative N/A

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 60.3

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

   11. Simplified 60.3%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. div-inv N/A

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

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

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

      11. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 94.8%

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

   8. Taylor expanded in l around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.3

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

   10. Simplified 85.3%

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

Alternative 14: 74.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l J) (fma -0.25 (* K K) 2.0) U)
   (fma l (* J (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(l, (J * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(l * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 57.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

       10. *-commutative N/A

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 60.3

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

   11. Simplified 60.3%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.7%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 87.5%

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

   6. Taylor expanded in K around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 82.4

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

   8. Simplified 82.4%

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

Alternative 15: 71.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(\ell \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -130000000000:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* l l))))
   (if (<= l -130000000000.0)
     (* 0.3333333333333333 (* J t_0))
     (if (<= l 2.45e+20)
       (fma (* 2.0 l) J U)
       (* J (* 0.3333333333333333 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (l * l);
	double tmp;
	if (l <= -130000000000.0) {
		tmp = 0.3333333333333333 * (J * t_0);
	} else if (l <= 2.45e+20) {
		tmp = fma((2.0 * l), J, U);
	} else {
		tmp = J * (0.3333333333333333 * t_0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(l * l))
	tmp = 0.0
	if (l <= -130000000000.0)
		tmp = Float64(0.3333333333333333 * Float64(J * t_0));
	elseif (l <= 2.45e+20)
		tmp = fma(Float64(2.0 * l), J, U);
	else
		tmp = Float64(J * Float64(0.3333333333333333 * t_0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -130000000000.0], N[(0.3333333333333333 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45e+20], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3e11

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

   5. Simplified 80.2%

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

   6. Taylor expanded in K around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 53.2

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

   8. Simplified 53.2%

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

   9. Taylor expanded in l around inf

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

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

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

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

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 57.8

          \[\leadsto 0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   11. Simplified 57.8%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -1.3e11 < l < 2.45e20

   1. Initial program 79.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 84.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 80.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

   10. Simplified 80.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

   if 2.45e20 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), U\right)} \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right), U\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right)}, U\right) \]

      7. *-lowering-*.f64 59.9

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right), U\right) \]

   8. Simplified 59.9%

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, U\right) \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{U}{J}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{U}{J}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto J \cdot \color{blue}{\mathsf{fma}\left(\ell, 2 + \frac{1}{3} \cdot {\ell}^{2}, \frac{U}{J}\right)} \]

       3. +-commutative N/A

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{3} \cdot {\ell}^{2} + 2}, \frac{U}{J}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{2}, 2\right)}, \frac{U}{J}\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right), \frac{U}{J}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\ell \cdot \ell}, 2\right), \frac{U}{J}\right) \]

       7. /-lowering-/.f64 61.3

          \[\leadsto J \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \color{blue}{\frac{U}{J}}\right) \]

   11. Simplified 61.3%

       \[\leadsto \color{blue}{J \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right), \frac{U}{J}\right)} \]

   12. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{3}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left({\ell}^{3} \cdot \frac{1}{3}\right)} \]

       3. cube-mult N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)} \cdot \frac{1}{3}\right) \]

       4. unpow2 N/A

          \[\leadsto J \cdot \left(\left(\ell \cdot \color{blue}{{\ell}^{2}}\right) \cdot \frac{1}{3}\right) \]

       5. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)} \]

       9. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \ell\right)\right) \]

       11. unpow3 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{{\ell}^{3}}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       13. cube-mult N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

       17. *-lowering-*.f64 64.5

           \[\leadsto J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   14. Simplified 64.5%

       \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 71.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -6000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (* l (* l l))))))
   (if (<= l -6000000000.0) t_0 (if (<= l 2.5e+20) (fma (* 2.0 l) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * (l * (l * l)));
	double tmp;
	if (l <= -6000000000.0) {
		tmp = t_0;
	} else if (l <= 2.5e+20) {
		tmp = fma((2.0 * l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -6000000000.0)
		tmp = t_0;
	elseif (l <= 2.5e+20)
		tmp = fma(Float64(2.0 * l), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6000000000.0], t$95$0, If[LessEqual[l, 2.5e+20], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -6e9 or 2.5e20 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), U\right)} \]

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}, U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{3} + 2\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)} + 2\right), U\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{3}, 2\right)}, U\right) \]

      7. *-lowering-*.f64 56.6

         \[\leadsto \mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.3333333333333333}, 2\right), U\right) \]

   8. Simplified 56.6%

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, U\right) \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(J \cdot {\ell}^{3}\right)} \]

       3. cube-mult N/A

          \[\leadsto \frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \left(J \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

       7. *-lowering-*.f64 61.2

          \[\leadsto 0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   11. Simplified 61.2%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -6e9 < l < 2.5e20

   1. Initial program 79.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 84.5%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 80.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

   10. Simplified 80.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -5500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= l -5500000000.0) t_0 (if (<= l 4.6e+21) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (l <= -5500000000.0) {
		tmp = t_0;
	} else if (l <= 4.6e+21) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (2.0d0 * l)
    if (l <= (-5500000000.0d0)) then
        tmp = t_0
    else if (l <= 4.6d+21) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (l <= -5500000000.0) {
		tmp = t_0;
	} else if (l <= 4.6e+21) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	tmp = 0
	if l <= -5500000000.0:
		tmp = t_0
	elif l <= 4.6e+21:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * l))
	tmp = 0.0
	if (l <= -5500000000.0)
		tmp = t_0;
	elseif (l <= 4.6e+21)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * l);
	tmp = 0.0;
	if (l <= -5500000000.0)
		tmp = t_0;
	elseif (l <= 4.6e+21)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5500000000.0], t$95$0, If[LessEqual[l, 4.6e+21], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.5e9 or 4.6e21 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

      5. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

      11. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
   6. Step-by-step derivation

   7. Simplified 70.9%

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 22.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

   10. Simplified 22.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \ell} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot 2\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

       5. *-lowering-*.f64 22.1

          \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

   13. Simplified 22.1%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

   if -5.5e9 < l < 4.6e21

   1. Initial program 79.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 72.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 53.9% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \ell, J, U\right) \end{array} \]

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(Float64(2.0 * l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 89.7%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)}, J, U\right) \]

   5. sinh-undef N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

   7. sinh-lowering-sinh.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}, J, U\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, J, U\right) \]

   11. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot \color{blue}{0.5}\right), J, U\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right)} \]

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]
6. Step-by-step derivation

7. Simplified 77.7%

   \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{1}, J, U\right) \]

8. Taylor expanded in l around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 51.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]

10. Simplified 51.4%

    \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \ell}, J, U\right) \]
11. Add Preprocessing

Alternative 19: 36.9% accurate, 330.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 89.7%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 37.4%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))