Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.9%

Time: 15.1s

Alternatives: 23

Speedup: 2.1×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

4. Applied egg-rr 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. Add Preprocessing

Alternative 2: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.86)
     (fma -0.25 (* J (* K (* l K))) (fma 2.0 (* l J) U))
     (if (<= t_0 -0.02)
       (fma (* l (* (cos (* K 0.5)) J)) 2.0 U)
       (fma (* (sinh l) J) 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.86) {
		tmp = fma(-0.25, (J * (K * (l * K))), fma(2.0, (l * J), U));
	} else if (t_0 <= -0.02) {
		tmp = fma((l * (cos((K * 0.5)) * J)), 2.0, U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.86)
		tmp = fma(-0.25, Float64(J * Float64(K * Float64(l * K))), fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(l * Float64(cos(Float64(K * 0.5)) * J)), 2.0, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.86], N[(-0.25 * N[(J * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.859999999999999987

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 63.5

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

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.5

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

   10. Applied egg-rr 78.5%

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

   if -0.859999999999999987 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 82.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 73.2

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

   5. Simplified 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 73.3

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

   7. Applied egg-rr 73.3%

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

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.86)
     (fma -0.25 (* J (* K (* l K))) (fma 2.0 (* l J) U))
     (if (<= t_0 -0.02)
       (fma (* (cos (* K 0.5)) (* 2.0 l)) J U)
       (fma (* (sinh l) J) 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.86) {
		tmp = fma(-0.25, (J * (K * (l * K))), fma(2.0, (l * J), U));
	} else if (t_0 <= -0.02) {
		tmp = fma((cos((K * 0.5)) * (2.0 * l)), J, U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.86)
		tmp = fma(-0.25, Float64(J * Float64(K * Float64(l * K))), fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * l)), J, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.86], N[(-0.25 * N[(J * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.859999999999999987

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 63.5

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

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.5

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

   10. Applied egg-rr 78.5%

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

   if -0.859999999999999987 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 82.0%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 73.3

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

   7. Simplified 73.3%

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

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.86)
     (fma -0.25 (* J (* K (* l K))) (fma 2.0 (* l J) U))
     (if (<= t_0 -0.02)
       (fma (cos (* K 0.5)) (* J (* 2.0 l)) U)
       (fma (* (sinh l) J) 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.86) {
		tmp = fma(-0.25, (J * (K * (l * K))), fma(2.0, (l * J), U));
	} else if (t_0 <= -0.02) {
		tmp = fma(cos((K * 0.5)), (J * (2.0 * l)), U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.86)
		tmp = fma(-0.25, Float64(J * Float64(K * Float64(l * K))), fma(2.0, Float64(l * J), U));
	elseif (t_0 <= -0.02)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(2.0 * l)), U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.86], N[(-0.25 * N[(J * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.859999999999999987

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 63.5

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

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.5

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

   10. Applied egg-rr 78.5%

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

   if -0.859999999999999987 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 82.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 73.2

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

   5. Simplified 73.2%

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

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.86:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot 0.16666666666666666, \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.98)
   (fma
    (* (cos (* K 0.5)) (* 2.0 (fma l (* (* l l) 0.16666666666666666) l)))
    J
    U)
   (fma (* (sinh l) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.98) {
		tmp = fma((cos((K * 0.5)) * (2.0 * fma(l, ((l * l) * 0.16666666666666666), l))), J, U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.98)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * fma(l, Float64(Float64(l * l) * 0.16666666666666666), l))), J, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.98], N[(N[(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[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.3%

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

   4. Applied egg-rr 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 89.3

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

   7. Simplified 89.3%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 87.2

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

   5. Simplified 87.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.0

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot 0.16666666666666666, \ell\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.98)
   (fma (* (cos (* K 0.5)) (* l (fma l (* l 0.3333333333333333) 2.0))) J U)
   (fma (* (sinh l) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.98) {
		tmp = fma((cos((K * 0.5)) * (l * fma(l, (l * 0.3333333333333333), 2.0))), J, U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.98)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0))), J, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.98], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.3%

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

   4. Applied egg-rr 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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 89.3

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

   7. Simplified 89.3%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 87.2

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

   5. Simplified 87.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.0

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 94.9%

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

Alternative 7: 92.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.98)
   (fma l (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (* l l) 2.0))) U)
   (fma (* (sinh l) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.98) {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(0.3333333333333333, (l * l), 2.0))), U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.98)
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0))), U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.98], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. 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 86.6%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 87.2

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

   5. Simplified 87.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.0

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 93.8%

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

Alternative 8: 87.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 56.7

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right) \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot -0.125}, 1\right) + U \]

   8. Simplified 56.7%

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot -0.125, 1\right)} + U \]

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 86.4%

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

Alternative 9: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-36)
   (fma (* (sinh l) J) 2.0 U)
   (+
    U
    (*
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma
         l
         (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
         0.3333333333333333)
        2.0)))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-36) {
		tmp = fma((sinh(l) * J), 2.0, U);
	} else {
		tmp = U + ((J * (l * fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos((K / 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-36)
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-36], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5.00000000000000004e-36

   1. Initial program 86.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 77.5

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

   5. Simplified 77.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 85.6

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

   7. Applied egg-rr 85.6%

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

   if 5.00000000000000004e-36 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.0

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

   5. Simplified 98.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1.2e-9)
   (fma (* (sinh l) J) 2.0 U)
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma (* l l) 0.016666666666666666 0.3333333333333333)
        2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1.2e-9) {
		tmp = fma((sinh(l) * J), 2.0, U);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1.2e-9)
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1.2e-9], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1.2e-9

   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 K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 77.6

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

   5. Simplified 77.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 85.9

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

   7. Applied egg-rr 85.9%

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

   if 1.2e-9 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 81.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.9

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

   5. Simplified 95.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 56.7

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right) \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot -0.125}, 1\right) + U \]

   8. Simplified 56.7%

      \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot -0.125, 1\right)} + U \]

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 83.7%

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

Alternative 12: 83.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (fma (* l (* l l)) 0.16666666666666666 l) (fma -0.25 (* K K) 2.0))
    J
    U)
   (fma
    J
    (*
     l
     (fma
      (* l l)
      (fma
       (* l l)
       (fma l (* l 0.0003968253968253968) 0.016666666666666666)
       0.3333333333333333)
      2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((fma((l * (l * l)), 0.16666666666666666, l) * fma(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma(J, (l * fma((l * l), fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), 0.16666666666666666, l) * fma(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(J, Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 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 91.4

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

   7. Simplified 91.4%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left({K}^{2} \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)} + 2 \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right), J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot {\ell}^{3} + \ell\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{{\ell}^{3} \cdot \frac{1}{6}} + \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\ell}^{3}, \frac{1}{6}, \ell\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{{\ell}^{2}}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot {\ell}^{2}}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{6}, \ell\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{6}, \ell\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      14. lower-*.f64 53.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

   10. Simplified 53.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)}, J, U\right) \]

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

Alternative 13: 92.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \ell \cdot \left(t\_0 \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.00162:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(J, e^{\ell} + \left(\ell + -1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* l (* t_0 (* J (fma l (* l 0.3333333333333333) 2.0))))))
   (if (<= l -2.25e+144)
     t_1
     (if (<= l -0.00162)
       (fma (* (sinh l) J) 2.0 U)
       (if (<= l 1.12e-13)
         (fma (* t_0 (* 2.0 l)) J U)
         (if (<= l 9.5e+100) (fma J (+ (exp l) (+ l -1.0)) U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = l * (t_0 * (J * fma(l, (l * 0.3333333333333333), 2.0)));
	double tmp;
	if (l <= -2.25e+144) {
		tmp = t_1;
	} else if (l <= -0.00162) {
		tmp = fma((sinh(l) * J), 2.0, U);
	} else if (l <= 1.12e-13) {
		tmp = fma((t_0 * (2.0 * l)), J, U);
	} else if (l <= 9.5e+100) {
		tmp = fma(J, (exp(l) + (l + -1.0)), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(l * Float64(t_0 * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))))
	tmp = 0.0
	if (l <= -2.25e+144)
		tmp = t_1;
	elseif (l <= -0.00162)
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	elseif (l <= 1.12e-13)
		tmp = fma(Float64(t_0 * Float64(2.0 * l)), J, U);
	elseif (l <= 9.5e+100)
		tmp = fma(J, Float64(exp(l) + Float64(l + -1.0)), U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(l * N[(t$95$0 * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.25e+144], t$95$1, If[LessEqual[l, -0.00162], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[l, 1.12e-13], N[(N[(t$95$0 * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 9.5e+100], N[(J * N[(N[Exp[l], $MachinePrecision] + N[(l + -1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.24999999999999984e144 or 9.4999999999999995e100 < l

   1. Initial program 100.0%

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

   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 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 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in J around inf

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

   10. Simplified 96.4%

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

   if -2.24999999999999984e144 < l < -0.0016199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 80.6

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

   5. Simplified 80.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. lift-sinh.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 80.6

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

   7. Applied egg-rr 80.6%

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

   if -0.0016199999999999999 < l < 1.12e-13

   1. Initial program 70.0%

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

   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

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

      1. lower-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 1.12e-13 < l < 9.4999999999999995e100

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in l around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 90.9

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

   8. Simplified 90.9%

      \[\leadsto \mathsf{fma}\left(J, e^{\ell} - \color{blue}{\left(1 - \ell\right)}, U\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+144}:\\ \;\;\;\;\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00162:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(J, e^{\ell} + \left(\ell + -1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (fma (* l (* l l)) 0.16666666666666666 l) (fma -0.25 (* K K) 2.0))
    J
    U)
   (fma
    J
    (*
     l
     (fma (* l l) (fma (* l l) 0.016666666666666666 0.3333333333333333) 2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((fma((l * (l * l)), 0.16666666666666666, l) * fma(-0.25, (K * K), 2.0)), J, U);
	} else {
		tmp = fma(J, (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(fma(Float64(l * Float64(l * l)), 0.16666666666666666, l) * fma(-0.25, Float64(K * K), 2.0)), J, U);
	else
		tmp = fma(J, Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 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))) < -0.0200000000000000004

   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 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 91.4

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

   7. Simplified 91.4%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left({K}^{2} \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)}, J, U\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right)} + 2 \cdot \left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right), J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell + \frac{1}{6} \cdot {\ell}^{3}\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot {\ell}^{3} + \ell\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{{\ell}^{3} \cdot \frac{1}{6}} + \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\ell}^{3}, \frac{1}{6}, \ell\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{{\ell}^{2}}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\ell \cdot {\ell}^{2}}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}, \frac{1}{6}, \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right), J, U\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{6}, \ell\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}, J, U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{6}, \ell\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

      14. lower-*.f64 53.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right), J, U\right) \]

   10. Simplified 53.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), 0.16666666666666666, \ell\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)}, J, U\right) \]

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

   1. Initial program 85.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.0

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

   8. Simplified 89.0%

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

Alternative 15: 76.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\ell \cdot K\right) \cdot \left(K \cdot J\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.31)
   (fma -0.25 (* (* l K) (* K J)) (fma 2.0 (* l J) U))
   (fma
    J
    (*
     l
     (fma (* l l) (fma (* l l) 0.016666666666666666 0.3333333333333333) 2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.31) {
		tmp = fma(-0.25, ((l * K) * (K * J)), fma(2.0, (l * J), U));
	} else {
		tmp = fma(J, (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.31)
		tmp = fma(-0.25, Float64(Float64(l * K) * Float64(K * J)), fma(2.0, Float64(l * J), U));
	else
		tmp = fma(J, Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.31], N[(-0.25 * N[(N[(l * K), $MachinePrecision] * N[(K * J), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 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))) < -0.309999999999999998

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 42.9

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

   8. Simplified 42.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 53.4

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

   10. Applied egg-rr 53.4%

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

   if -0.309999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.1

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

   8. Simplified 87.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\ell \cdot K\right) \cdot \left(K \cdot J\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\ell \cdot K\right) \cdot \left(K \cdot J\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.31)
   (fma -0.25 (* (* l K) (* K J)) (fma 2.0 (* l J) U))
   (fma J (* l (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.31) {
		tmp = fma(-0.25, ((l * K) * (K * J)), fma(2.0, (l * J), U));
	} else {
		tmp = fma(J, (l * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.31)
		tmp = fma(-0.25, Float64(Float64(l * K) * Float64(K * J)), fma(2.0, Float64(l * J), U));
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.31], N[(-0.25 * N[(N[(l * K), $MachinePrecision] * N[(K * J), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.309999999999999998

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 42.9

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

   8. Simplified 42.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 53.4

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

   10. Applied egg-rr 53.4%

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

   if -0.309999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 80.2

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

   8. Simplified 80.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\ell \cdot K\right) \cdot \left(K \cdot J\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.31)
   (fma -0.25 (* J (* K (* l K))) (fma 2.0 (* l J) U))
   (fma J (* l (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.31) {
		tmp = fma(-0.25, (J * (K * (l * K))), fma(2.0, (l * J), U));
	} else {
		tmp = fma(J, (l * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.31)
		tmp = fma(-0.25, Float64(J * Float64(K * Float64(l * K))), fma(2.0, Float64(l * J), U));
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.31], N[(-0.25 * N[(J * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision] + U), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.309999999999999998

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 42.9

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

   8. Simplified 42.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 51.3

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

   10. Applied egg-rr 51.3%

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

   if -0.309999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 80.2

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

   8. Simplified 80.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(-0.25, J \cdot \left(K \cdot \left(\ell \cdot K\right)\right), \mathsf{fma}\left(2, \ell \cdot J, U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 66.8

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

   5. Simplified 66.8%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 50.1

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

   11. Simplified 50.1%

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

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

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 81.7

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

   8. Simplified 81.7%

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

4. Final simplification 74.8%

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

Alternative 19: 70.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\ell \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e+294)
   (fma J (* l (fma l (* l 0.3333333333333333) 2.0)) U)
   (* -0.25 (* l (* J (* K K))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e+294) {
		tmp = fma(J, (l * fma(l, (l * 0.3333333333333333), 2.0)), U);
	} else {
		tmp = -0.25 * (l * (J * (K * K)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e+294)
		tmp = fma(J, Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	else
		tmp = Float64(-0.25 * Float64(l * Float64(J * Float64(K * K))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e+294], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(-0.25 * N[(l * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.9999999999999999e294

   1. Initial program 85.2%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 72.3

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

   8. Simplified 72.3%

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

   if 4.9999999999999999e294 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 50.0

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

   8. Simplified 50.0%

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

   9. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\ell \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\ell \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e+294) (fma J (* 2.0 l) U) (* -0.25 (* l (* J (* K K))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e+294) {
		tmp = fma(J, (2.0 * l), U);
	} else {
		tmp = -0.25 * (l * (J * (K * K)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e+294)
		tmp = fma(J, Float64(2.0 * l), U);
	else
		tmp = Float64(-0.25 * Float64(l * Float64(J * Float64(K * K))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e+294], N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], N[(-0.25 * N[(l * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.9999999999999999e294

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) + U} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}, U\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

      4. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}, U\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

      6. lower-neg.f64 74.9

         \[\leadsto \mathsf{fma}\left(J, e^{\ell} - e^{\color{blue}{-\ell}}, U\right) \]

   5. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 52.7

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

   8. Simplified 52.7%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

   if 4.9999999999999999e294 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 52.3

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \left(2 \cdot \left(J \cdot \ell\right) + U\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \color{blue}{\left(U + 2 \cdot \left(J \cdot \ell\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, J \cdot \left({K}^{2} \cdot \ell\right), U + 2 \cdot \left(J \cdot \ell\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{J \cdot \left({K}^{2} \cdot \ell\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \color{blue}{\left(K \cdot K\right)}\right), U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \color{blue}{\left(K \cdot K\right)}\right), U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \color{blue}{2 \cdot \left(J \cdot \ell\right) + U}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)}\right) \]

      12. lower-*.f64 50.0

          \[\leadsto \mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right)\right) \]

   8. Simplified 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]

   9. Taylor expanded in K around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot {K}^{2}\right) \cdot \ell\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\ell \cdot \left(J \cdot {K}^{2}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\ell \cdot \left(J \cdot {K}^{2}\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(\ell \cdot \color{blue}{\left(J \cdot {K}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(\ell \cdot \left(J \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

       7. lower-*.f64 100.0

          \[\leadsto -0.25 \cdot \left(\ell \cdot \left(J \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\ell \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 58.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right)\\ \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J (fma -0.25 (* K K) 2.0)))))
   (if (<= l -7.6e+72) t_0 (if (<= l 3e+27) (fma J (* 2.0 l) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * fma(-0.25, (K * K), 2.0));
	double tmp;
	if (l <= -7.6e+72) {
		tmp = t_0;
	} else if (l <= 3e+27) {
		tmp = fma(J, (2.0 * l), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * fma(-0.25, Float64(K * K), 2.0)))
	tmp = 0.0
	if (l <= -7.6e+72)
		tmp = t_0;
	elseif (l <= 3e+27)
		tmp = fma(J, Float64(2.0 * l), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.6e+72], t$95$0, If[LessEqual[l, 3e+27], N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -7.60000000000000012e72 or 2.99999999999999976e27 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 29.1

          \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \color{blue}{\left(\ell \cdot 2\right)}, U\right) \]

   5. Simplified 29.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell \cdot 2\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \left(2 \cdot \left(J \cdot \ell\right) + U\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \color{blue}{\left(U + 2 \cdot \left(J \cdot \ell\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, J \cdot \left({K}^{2} \cdot \ell\right), U + 2 \cdot \left(J \cdot \ell\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{J \cdot \left({K}^{2} \cdot \ell\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \color{blue}{\left(K \cdot K\right)}\right), U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \color{blue}{\left(K \cdot K\right)}\right), U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \color{blue}{2 \cdot \left(J \cdot \ell\right) + U}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \color{blue}{\mathsf{fma}\left(2, J \cdot \ell, U\right)}\right) \]

      12. lower-*.f64 28.6

          \[\leadsto \mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, \color{blue}{J \cdot \ell}, U\right)\right) \]

   8. Simplified 28.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, J \cdot \left(\ell \cdot \left(K \cdot K\right)\right), \mathsf{fma}\left(2, J \cdot \ell, U\right)\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot J + \left(2 \cdot \ell\right) \cdot J} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot J\right)} + \left(2 \cdot \ell\right) \cdot J \]

       3. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + \left(2 \cdot \ell\right) \cdot J \]

       4. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot {K}^{2}\right) \cdot \ell\right)} + \left(2 \cdot \ell\right) \cdot J \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \ell} + \left(2 \cdot \ell\right) \cdot J \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} + \left(2 \cdot \ell\right) \cdot J \]

       7. *-commutative N/A

          \[\leadsto \ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) + \color{blue}{\left(\ell \cdot 2\right)} \cdot J \]

       8. associate-*r* N/A

          \[\leadsto \ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) + \color{blue}{\ell \cdot \left(2 \cdot J\right)} \]

       9. distribute-lft-in N/A

          \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)} \]

       11. *-commutative N/A

           \[\leadsto \ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left({K}^{2} \cdot J\right)} + 2 \cdot J\right) \]

       12. associate-*r* N/A

           \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot J} + 2 \cdot J\right) \]

       13. distribute-rgt-out N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right)} \]

       15. lower-fma.f64 N/A

           \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right)\right) \]

       17. lower-*.f64 36.1

           \[\leadsto \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.25, \color{blue}{K \cdot K}, 2\right)\right) \]

   11. Simplified 36.1%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right)} \]

   if -7.60000000000000012e72 < l < 2.99999999999999976e27

   1. Initial program 73.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) + U} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}, U\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

      4. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}, U\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

      6. lower-neg.f64 71.0

         \[\leadsto \mathsf{fma}\left(J, e^{\ell} - e^{\color{blue}{-\ell}}, U\right) \]

   5. Simplified 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 76.4

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

   8. Simplified 76.4%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 53.1% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, 2 \cdot \ell, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma J (* 2.0 l) U))

C

double code(double J, double l, double K, double U) {
	return fma(J, (2.0 * l), U);
}

Julia

function code(J, l, K, U)
	return fma(J, Float64(2.0 * l), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 85.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) + U} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}, U\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

   4. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}, U\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

   6. lower-neg.f64 74.3

      \[\leadsto \mathsf{fma}\left(J, e^{\ell} - e^{\color{blue}{-\ell}}, U\right) \]

5. Simplified 74.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)} \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
7. Step-by-step derivation

   1. lower-*.f64 52.3

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

8. Simplified 52.3%

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
9. Add Preprocessing

Alternative 23: 19.1% accurate, 30.0× speedup

Math

\[\begin{array}{l} \\ J \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* J (* 2.0 l)))

C

double code(double J, double l, double K, double U) {
	return J * (2.0 * l);
}

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 * (2.0d0 * l)
end function

Java

public static double code(double J, double l, double K, double U) {
	return J * (2.0 * l);
}

Python

def code(J, l, K, U):
	return J * (2.0 * l)

Julia

function code(J, l, K, U)
	return Float64(J * Float64(2.0 * l))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = J * (2.0 * l);
end

Wolfram

code[J_, l_, K_, U_] := N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) + U} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}, U\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

   4. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}, U\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J, e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}, U\right) \]

   6. lower-neg.f64 74.3

      \[\leadsto \mathsf{fma}\left(J, e^{\ell} - e^{\color{blue}{-\ell}}, U\right) \]

5. Simplified 74.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)} \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]
7. Step-by-step derivation

   1. lower-*.f64 52.3

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

8. Simplified 52.3%

   \[\leadsto \mathsf{fma}\left(J, \color{blue}{2 \cdot \ell}, U\right) \]

9. Taylor expanded in J around inf

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

    3. *-commutative N/A

       \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

    5. *-commutative N/A

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

    6. lower-*.f64 19.5

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

11. Simplified 19.5%

    \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

12. Final simplification 19.5%

    \[\leadsto J \cdot \left(2 \cdot \ell\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))