Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 13.5s

Alternatives: 21

Speedup: 1.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

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. Final simplification 100.0%

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

Alternative 2: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (*
     (cos (* K 0.5))
     (*
      l
      (fma
       (* l l)
       (fma
        (* l l)
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        0.3333333333333333)
       2.0)))
    J
    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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (l * fma((l * l), fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0))), J, 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0))), J, 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * 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] * J + 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))) < 0.96999999999999997

   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. Step-by-step derivation

   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 l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 97.6

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   7. Simplified 97.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)} \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (fma
     (* l (* l l))
     (fma (* l l) (* l 0.016666666666666666) 0.3333333333333333)
     2.0)
    (* (cos (* K 0.5)) (* l J))
    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)) <= 0.97) {
		tmp = fma(fma((l * (l * l)), fma((l * l), (l * 0.016666666666666666), 0.3333333333333333), 2.0), (cos((K * 0.5)) * (l * J)), 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)) <= 0.97)
		tmp = fma(fma(Float64(l * Float64(l * l)), fma(Float64(l * l), Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0), Float64(cos(Float64(K * 0.5)) * Float64(l * J)), 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], 0.97], N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * J), $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))) < 0.96999999999999997

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in J around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 97.4

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

   8. Simplified 97.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

   10. Applied egg-rr 97.4%

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

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      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 <= 0.97) {
		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 <= 0.97)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(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, 0.97], N[(U + N[(t$95$0 * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 0.96999999999999997

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 96.7

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

   5. Simplified 96.7%

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

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 98.2%

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

Alternative 5: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot 0.16666666666666666, \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma
    (* (cos (* K 0.5)) (* 2.0 (fma l (* (* l l) 0.16666666666666666) l)))
    J
    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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (2.0 * fma(l, ((l * l) * 0.16666666666666666), l))), J, 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * fma(l, Float64(Float64(l * l) * 0.16666666666666666), l))), J, 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(l * N[(N[(l * l), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   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. Step-by-step derivation

   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 l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.9

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

   7. Simplified 91.9%

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

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot 0.16666666666666666, \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma (* (cos (* K 0.5)) (* l (fma 0.3333333333333333 (* l l) 2.0))) J 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)) <= 0.97) {
		tmp = fma((cos((K * 0.5)) * (l * fma(0.3333333333333333, (l * l), 2.0))), J, 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)) <= 0.97)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(0.3333333333333333, Float64(l * l), 2.0))), J, 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], 0.97], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < 0.96999999999999997

   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. Step-by-step derivation

   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 l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 91.9

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

   7. Simplified 91.9%

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

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma l (* (cos (* K 0.5)) (* J (fma l (* l 0.3333333333333333) 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)) <= 0.97) {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(l, (l * 0.3333333333333333), 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)) <= 0.97)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * fma(l, Float64(l * 0.3333333333333333), 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], 0.97], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(l * 0.3333333333333333), $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))) < 0.96999999999999997

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 90.4%

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

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

   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. Step-by-step derivation

   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 99.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma l (* (cos (* K 0.5)) (* 0.3333333333333333 (* J (* l l)))) 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)) <= -0.01) {
		tmp = fma(l, (cos((K * 0.5)) * (0.3333333333333333 * (J * (l * l)))), 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)) <= -0.01)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(0.3333333333333333 * Float64(J * Float64(l * l)))), 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], -0.01], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * N[(J * N[(l * l), $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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 89.4%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 93.5

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

   9. Applied egg-rr 93.5%

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

4. Final simplification 91.1%

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

Alternative 9: 88.3% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma l (* (* 0.3333333333333333 (* J (* l l))) (fma K (* K -0.125) 1.0)) 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)) <= -0.01) {
		tmp = fma(l, ((0.3333333333333333 * (J * (l * l))) * fma(K, (K * -0.125), 1.0)), 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)) <= -0.01)
		tmp = fma(l, Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) * fma(K, Float64(K * -0.125), 1.0)), 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], -0.01], N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 89.4%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 76.6

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

   11. Simplified 76.6%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

      1. lift-sinh.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 93.5

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

   9. Applied egg-rr 93.5%

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

4. Final simplification 89.4%

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

Alternative 10: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(t\_1, \mathsf{fma}\left(t\_1, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* l (* l l)))) (t_1 (* t_0 t_0)))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma
      l
      (* (* 0.3333333333333333 (* J (* l l))) (fma K (* K -0.125) 1.0))
      U)
     (fma
      J
      (* l (fma t_1 (fma t_1 0.016666666666666666 0.3333333333333333) 2.0))
      U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (l * (l * l));
	double t_1 = t_0 * t_0;
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(l, ((0.3333333333333333 * (J * (l * l))) * fma(K, (K * -0.125), 1.0)), U);
	} else {
		tmp = fma(J, (l * fma(t_1, fma(t_1, 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(l * Float64(l * l)))
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(l, Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) * fma(K, Float64(K * -0.125), 1.0)), U);
	else
		tmp = fma(J, Float64(l * fma(t_1, fma(t_1, 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * N[(t$95$1 * N[(t$95$1 * 0.016666666666666666 + 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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 89.4%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 76.6

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

   11. Simplified 76.6%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in J around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 94.0

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

   8. Simplified 94.0%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Simplified 91.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{fma}\left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right), 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma l (* (* 0.3333333333333333 (* J (* l l))) (fma K (* K -0.125) 1.0)) U)
   (fma
    (*
     l
     (fma
      (* l l)
      (fma
       (* l l)
       (fma 0.0003968253968253968 (* 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)) <= -0.01) {
		tmp = fma(l, ((0.3333333333333333 * (J * (l * l))) * fma(K, (K * -0.125), 1.0)), U);
	} else {
		tmp = fma((l * fma((l * l), fma((l * l), fma(0.0003968253968253968, (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)) <= -0.01)
		tmp = fma(l, Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) * fma(K, Float64(K * -0.125), 1.0)), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(0.0003968253968253968, 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], -0.01], N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(0.0003968253968253968 * N[(l * l), $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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 89.4%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 76.6

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

   11. Simplified 76.6%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-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. lower-*.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. lower-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. lower-*.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. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.0

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

   10. Simplified 89.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma l (* (* 0.3333333333333333 (* J (* l l))) (fma K (* K -0.125) 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)) <= -0.01) {
		tmp = fma(l, ((0.3333333333333333 * (J * (l * l))) * fma(K, (K * -0.125), 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)) <= -0.01)
		tmp = fma(l, Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) * fma(K, Float64(K * -0.125), 1.0)), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Simplified 89.4%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 76.6

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

   11. Simplified 76.6%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

   8. Taylor expanded in l around 0

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

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

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   10. Simplified 86.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell, \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* (fma K (* K -0.125) 1.0) (* 2.0 l)) J U)
   (fma
    (*
     l
     (fma (* l l) (fma l (* l 0.016666666666666666) 0.3333333333333333) 2.0))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((fma(K, (K * -0.125), 1.0) * (2.0 * l)), J, U);
	} else {
		tmp = fma((l * fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(fma(K, Float64(K * -0.125), 1.0) * Float64(2.0 * l)), J, U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 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))) < -0.0100000000000000002

   1. Initial program 95.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.0

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

   5. Simplified 61.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.9

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

   8. Simplified 70.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, K \cdot K, 1\right)}, J \cdot \left(\ell \cdot 2\right), U\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 70.9

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 70.9

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

   10. Applied egg-rr 70.9%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

   8. Taylor expanded in l around 0

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

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

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   10. Simplified 86.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* (fma K (* K -0.125) 1.0) (* 2.0 l)) J U)
   (fma (* l (fma l (* l 0.3333333333333333) 2.0)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((fma(K, (K * -0.125), 1.0) * (2.0 * l)), J, U);
	} else {
		tmp = fma((l * fma(l, (l * 0.3333333333333333), 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(fma(K, Float64(K * -0.125), 1.0) * Float64(2.0 * l)), J, U);
	else
		tmp = fma(Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.0

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

   5. Simplified 61.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.9

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

   8. Simplified 70.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, K \cdot K, 1\right)}, J \cdot \left(\ell \cdot 2\right), U\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 70.9

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 70.9

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

   10. Applied egg-rr 70.9%

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

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

   1. Initial program 83.5%

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

   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 93.5%

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

   8. Taylor expanded in l around 0

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

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

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.3

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

   10. Simplified 83.3%

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

4. Final simplification 80.3%

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

Alternative 15: 22.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = U * (U * U);
	} else {
		tmp = 2.0 * (l * J);
	}
	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) :: tmp
    if (cos((k / 2.0d0)) <= 0.055d0) then
        tmp = u * (u * u)
    else
        tmp = 2.0d0 * (l * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.055) {
		tmp = U * (U * U);
	} else {
		tmp = 2.0 * (l * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.055:
		tmp = U * (U * U)
	else:
		tmp = 2.0 * (l * J)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.055)
		tmp = U * (U * U);
	else
		tmp = 2.0 * (l * J);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.3

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

   5. Simplified 59.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 34.8

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

   8. Simplified 34.8%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. swap-sqr N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-neg.f64 20.0

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

   10. Applied egg-rr 20.0%

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

   11. Taylor expanded in l around 0

       \[\leadsto \color{blue}{{U}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{U \cdot \left(U \cdot U\right)} \]

       2. unpow2 N/A

          \[\leadsto U \cdot \color{blue}{{U}^{2}} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{U \cdot {U}^{2}} \]

       4. unpow2 N/A

          \[\leadsto U \cdot \color{blue}{\left(U \cdot U\right)} \]

       5. lower-*.f64 15.5

          \[\leadsto U \cdot \color{blue}{\left(U \cdot U\right)} \]

   13. Simplified 15.5%

       \[\leadsto \color{blue}{U \cdot \left(U \cdot U\right)} \]

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

   1. Initial program 83.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 62.3

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

   8. Simplified 62.3%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 29.3

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

   11. Simplified 29.3%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;U \cdot \left(U \cdot U\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.4% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

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 79.1%

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

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. lower-*.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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 71.5

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

10. Simplified 71.5%

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

Alternative 17: 54.3% 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 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Simplified 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Simplified 55.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 55.3

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

10. Applied egg-rr 55.3%

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

11. Final simplification 55.3%

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

Alternative 18: 54.3% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Simplified 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Simplified 55.3%

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

9. Final simplification 55.3%

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

Alternative 19: 19.8% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

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 = 2.0d0 * (l * j)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Simplified 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Simplified 55.3%

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

9. Taylor expanded in l around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 22.2

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

11. Simplified 22.2%

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

12. Final simplification 22.2%

    \[\leadsto 2 \cdot \left(\ell \cdot J\right) \]
13. Add Preprocessing

Alternative 20: 3.0% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot J \end{array} \]

FPCore

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

C

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

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 = 2.0d0 * j
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Simplified 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Simplified 55.3%

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

9. Taylor expanded in J around inf

   \[\leadsto \color{blue}{2 \cdot J} \]
10. Step-by-step derivation

    1. lower-*.f64 3.1

       \[\leadsto \color{blue}{2 \cdot J} \]

11. Simplified 3.1%

    \[\leadsto \color{blue}{2 \cdot J} \]
12. Add Preprocessing

Alternative 21: 3.0% accurate, 110.0× speedup

Math

\[\begin{array}{l} \\ -J \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (- J))

C

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

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
end function

Java

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

Python

def code(J, l, K, U):
	return -J

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := (-J)

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 66.3

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

5. Simplified 66.3%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 55.3

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

8. Simplified 55.3%

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

9. Taylor expanded in J around inf

   \[\leadsto \color{blue}{2 \cdot J} \]
10. Step-by-step derivation

    1. lower-*.f64 3.1

       \[\leadsto \color{blue}{2 \cdot J} \]

11. Simplified 3.1%

    \[\leadsto \color{blue}{2 \cdot J} \]

12. Taylor expanded in J around -inf

    \[\leadsto \color{blue}{-1 \cdot J} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(J\right)} \]

    2. lower-neg.f64 3.1

       \[\leadsto \color{blue}{-J} \]

14. Simplified 3.1%

    \[\leadsto \color{blue}{-J} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))