Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 100.0%

Time: 13.0s

Alternatives: 21

Speedup: 2.4×

• 47.4% 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: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= t_0 -0.0005)
     (fma (* J l) (fma -0.25 (* K K) 2.0) U)
     (if (<= t_0 2e-13)
       (fma J (* l 2.0) U)
       (fma (* l (fma (* K K) -0.25 2.0)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = fma((J * l), fma(-0.25, (K * K), 2.0), U);
	} else if (t_0 <= 2e-13) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = fma(Float64(J * l), fma(-0.25, Float64(K * K), 2.0), U);
	elseif (t_0 <= 2e-13)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0005], N[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5.0000000000000001e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 35.4%

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

   if -5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-13

   1. Initial program 72.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 89.0%

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

   if 2.0000000000000001e-13 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 35.8%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 43.4%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l))))
        (t_1 (fma (* J l) (fma -0.25 (* K K) 2.0) U)))
   (if (<= t_0 -0.0005) t_1 (if (<= t_0 2e-13) (fma J (* l 2.0) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = fma((J * l), fma(-0.25, (K * K), 2.0), U);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = fma(Float64(J * l), fma(-0.25, Float64(K * K), 2.0), U)
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * l), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0005], t$95$1, If[LessEqual[t$95$0, 2e-13], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5.0000000000000001e-4 or 2.0000000000000001e-13 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 26.0

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 38.2%

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

   if -5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-13

   1. Initial program 72.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 89.0%

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

Alternative 4: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.999)
   (fma
    (*
     (* J (cos (* K 0.5)))
     (fma
      (fma
       (* l l)
       (fma (* l l) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      (* l (* l l))
      l))
    2.0
    U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.999) {
		tmp = fma(((J * cos((K * 0.5))) * fma(fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (l * (l * l)), l)), 2.0, U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.999)
		tmp = fma(Float64(Float64(J * cos(Float64(K * 0.5))) * fma(fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(l * Float64(l * l)), l)), 2.0, U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.998999999999999999

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 93.6%

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

   if 0.998999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.999)
   (fma
    (*
     (* J (cos (* K 0.5)))
     (fma
      (* l l)
      (* l (fma (* l l) 0.008333333333333333 0.16666666666666666))
      l))
    2.0
    U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.999) {
		tmp = fma(((J * cos((K * 0.5))) * fma((l * l), (l * fma((l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.999)
		tmp = fma(Float64(Float64(J * cos(Float64(K * 0.5))) * fma(Float64(l * l), Float64(l * fma(Float64(l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.998999999999999999

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

   if 0.998999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.999)
   (fma
    (cos (* K 0.5))
    (*
     l
     (*
      J
      (fma (* l l) (fma l (* l 0.016666666666666666) 0.3333333333333333) 2.0)))
    U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.999) {
		tmp = fma(cos((K * 0.5)), (l * (J * fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.999)
		tmp = fma(cos(Float64(K * 0.5)), Float64(l * Float64(J * fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.998999999999999999

   1. Initial program 87.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 90.9

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied rewrites 91.7%

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

   if 0.998999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \cos \left(K \cdot 0.5\right), \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.4)
   (fma (* J (cos (* K 0.5))) (* l (fma l (* l 0.3333333333333333) 2.0)) U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.4) {
		tmp = fma((J * cos((K * 0.5))), (l * fma(l, (l * 0.3333333333333333), 2.0)), U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.4)
		tmp = fma(Float64(J * cos(Float64(K * 0.5))), Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.4], N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.40000000000000002

   1. Initial program 90.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if 0.40000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 96.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \cos \left(K \cdot 0.5\right), \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma l (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (* l l) 2.0))) U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(0.3333333333333333, (l * l), 2.0))), U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		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) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], 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] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 91.4%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 87.2%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 95.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma
    (fma K (* K -0.125) 1.0)
    (* l (* l (* J (* l (* (* l l) 0.016666666666666666)))))
    U)
   (fma (* (sinh l) (* J 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (l * (J * (l * ((l * l) * 0.016666666666666666))))), U);
	} else {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(l * Float64(J * Float64(l * Float64(Float64(l * l) * 0.016666666666666666))))), U);
	else
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 91.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 92.7

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

   7. Applied rewrites 92.7%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 58.1

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

   10. Applied rewrites 58.1%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot \frac{-1}{8}, 1\right), \ell \cdot \left(\frac{1}{60} \cdot \color{blue}{\left(J \cdot {\ell}^{4}\right)}\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 67.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 95.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.205:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right) \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.205)
   (fma
    (fma K (* K -0.125) 1.0)
    (* l (* l (* J (* l (* (* l l) 0.016666666666666666)))))
    U)
   (fma
    (*
     (fma
      (fma
       (* l l)
       (fma (* l l) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      (* l (* l l))
      l)
     (* J 1.0))
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.205) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (l * (J * (l * ((l * l) * 0.016666666666666666))))), U);
	} else {
		tmp = fma((fma(fma((l * l), fma((l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (l * (l * l)), l) * (J * 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.205)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(l * Float64(J * Float64(l * Float64(Float64(l * l) * 0.016666666666666666))))), U);
	else
		tmp = fma(Float64(fma(fma(Float64(l * l), fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(l * Float64(l * l)), l) * Float64(J * 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.205], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.204999999999999988

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.8

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

   10. Applied rewrites 59.8%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot \frac{-1}{8}, 1\right), \ell \cdot \left(\frac{1}{60} \cdot \color{blue}{\left(J \cdot {\ell}^{4}\right)}\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 68.5%

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

   if -0.204999999999999988 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 86.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.205:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \ell \cdot \left(\ell \cdot \ell\right), \ell\right) \cdot \left(J \cdot 1\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.205:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.205)
   (fma
    (fma K (* K -0.125) 1.0)
    (* l (* l (* J (* l (* (* l l) 0.016666666666666666)))))
    U)
   (fma
    (*
     (* J 1.0)
     (fma
      (* l l)
      (* l (fma (* l l) 0.008333333333333333 0.16666666666666666))
      l))
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.205) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (l * (J * (l * ((l * l) * 0.016666666666666666))))), U);
	} else {
		tmp = fma(((J * 1.0) * fma((l * l), (l * fma((l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.205)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(l * Float64(J * Float64(l * Float64(Float64(l * l) * 0.016666666666666666))))), U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(l * l), Float64(l * fma(Float64(l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.205], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.204999999999999988

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.8

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

   10. Applied rewrites 59.8%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot \frac{-1}{8}, 1\right), \ell \cdot \left(\frac{1}{60} \cdot \color{blue}{\left(J \cdot {\ell}^{4}\right)}\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 68.5%

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

   if -0.204999999999999988 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 85.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.008333333333333333, 0.16666666666666666\right) \cdot \ell, \ell\right), 2, U\right) \]

   10. Applied rewrites 85.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.205:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma
    (fma K (* K -0.125) 1.0)
    (* l (* J (fma l (* l 0.3333333333333333) 2.0)))
    U)
   (fma
    (*
     (* J 1.0)
     (fma
      (* l l)
      (* l (fma (* l l) 0.008333333333333333 0.16666666666666666))
      l))
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.15) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma(((J * 1.0) * fma((l * l), (l * fma((l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.15)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(Float64(l * l), Float64(l * fma(Float64(l * l), 0.008333333333333333, 0.16666666666666666)), l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.15], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.6

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

   10. Applied rewrites 59.6%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 59.6

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

   13. Applied rewrites 59.6%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 86.2

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.008333333333333333, 0.16666666666666666\right) \cdot \ell, \ell\right), 2, U\right) \]

   10. Applied rewrites 86.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.008333333333333333, 0.16666666666666666\right), \ell\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma
    (fma K (* K -0.125) 1.0)
    (* l (* J (fma l (* l 0.3333333333333333) 2.0)))
    U)
   (fma
    1.0
    (*
     l
     (*
      J
      (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)) <= -0.15) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma(1.0, (l * (J * 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)) <= -0.15)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(1.0, Float64(l * Float64(J * fma(l, Float64(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], -0.15], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(1.0 * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 59.6

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

   10. Applied rewrites 59.6%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 59.6

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

   13. Applied rewrites 59.6%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

   13. Applied rewrites 84.3%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma (* l (fma (* K K) -0.25 2.0)) J U)
   (fma
    1.0
    (*
     l
     (*
      J
      (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)) <= -0.15) {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	} else {
		tmp = fma(1.0, (l * (J * 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)) <= -0.15)
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	else
		tmp = fma(1.0, Float64(l * Float64(J * fma(l, Float64(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], -0.15], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(1.0 * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 55.4%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

   13. Applied rewrites 84.3%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot 0.16666666666666666, \ell\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma (* l (fma (* K K) -0.25 2.0)) J U)
   (fma (* (* J 1.0) (fma l (* (* l l) 0.16666666666666666) l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.15) {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	} else {
		tmp = fma(((J * 1.0) * fma(l, ((l * l) * 0.16666666666666666), l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.15)
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	else
		tmp = fma(Float64(Float64(J * 1.0) * fma(l, Float64(Float64(l * l) * 0.16666666666666666), l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.15], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * 1.0), $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 55.4%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.8

         \[\leadsto \mathsf{fma}\left(\left(J \cdot 1\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.16666666666666666, \ell\right), 2, U\right) \]

   10. Applied rewrites 80.8%

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

Alternative 17: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \ell \cdot \mathsf{fma}\left(\ell, J \cdot \left(\ell \cdot 0.3333333333333333\right), J \cdot 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma (* l (fma (* K K) -0.25 2.0)) J U)
   (fma 1.0 (* l (fma l (* J (* l 0.3333333333333333)) (* J 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.15) {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	} else {
		tmp = fma(1.0, (l * fma(l, (J * (l * 0.3333333333333333)), (J * 2.0))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.15)
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	else
		tmp = fma(1.0, Float64(l * fma(l, Float64(J * Float64(l * 0.3333333333333333)), Float64(J * 2.0))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.15], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(1.0 * N[(l * N[(l * N[(J * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 55.4%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 77.9%

       \[\leadsto \mathsf{fma}\left(1, \ell \cdot \mathsf{fma}\left(\ell, J \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333\right)}, J \cdot 2\right), U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 74.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.15)
   (fma (* l (fma (* K K) -0.25 2.0)) J U)
   (fma 1.0 (* l (* J (fma l (* l 0.3333333333333333) 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.15) {
		tmp = fma((l * fma((K * K), -0.25, 2.0)), J, U);
	} else {
		tmp = fma(1.0, (l * (J * fma(l, (l * 0.3333333333333333), 2.0))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.15)
		tmp = fma(Float64(l * fma(Float64(K * K), -0.25, 2.0)), J, U);
	else
		tmp = fma(1.0, Float64(l * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.15], N[(N[(l * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(1.0 * N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.149999999999999994

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 55.4%

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

   if -0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 77.9

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

   13. Applied rewrites 77.9%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 86.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{elif}\;\ell \leq 9000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \ell, 2, U\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.2)
   (fma (* (sinh l) (* J (fma -0.125 (* K K) 1.0))) 2.0 U)
   (if (<= l 9000000000.0)
     (fma (* (* J (cos (* K 0.5))) l) 2.0 U)
     (if (<= l 9.5e+120)
       (fma (* (sinh l) (* J 1.0)) 2.0 U)
       (fma
        (fma K (* K -0.125) 1.0)
        (* l (* J (fma l (* l 0.3333333333333333) 2.0)))
        U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.2) {
		tmp = fma((sinh(l) * (J * fma(-0.125, (K * K), 1.0))), 2.0, U);
	} else if (l <= 9000000000.0) {
		tmp = fma(((J * cos((K * 0.5))) * l), 2.0, U);
	} else if (l <= 9.5e+120) {
		tmp = fma((sinh(l) * (J * 1.0)), 2.0, U);
	} else {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * 0.3333333333333333), 2.0))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.2)
		tmp = fma(Float64(sinh(l) * Float64(J * fma(-0.125, Float64(K * K), 1.0))), 2.0, U);
	elseif (l <= 9000000000.0)
		tmp = fma(Float64(Float64(J * cos(Float64(K * 0.5))) * l), 2.0, U);
	elseif (l <= 9.5e+120)
		tmp = fma(Float64(sinh(l) * Float64(J * 1.0)), 2.0, U);
	else
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.2], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[l, 9000000000.0], N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[l, 9.5e+120], N[(N[(N[Sinh[l], $MachinePrecision] * N[(J * 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.2000000000000002

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -3.2000000000000002 < l < 9e9

   1. Initial program 72.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if 9e9 < l < 9.5e120

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 90.5%

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

   if 9.5e120 < 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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 88.6

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

   10. Applied rewrites 88.6%

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

   11. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 88.6

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

   13. Applied rewrites 88.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \color{blue}{\ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)}, U\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right), 2, U\right)\\ \mathbf{elif}\;\ell \leq 9000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \ell, 2, U\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \left(J \cdot 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 56.4% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 62.4

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

5. Applied rewrites 62.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 54.3%

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

Alternative 21: 20.4% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = l * (j * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 62.4

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

5. Applied rewrites 62.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 54.3%

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

9. Taylor expanded in J around inf

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

11. Applied rewrites 20.0%

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

Reproduce

herbie shell --seed 2024222 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))