Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 100.0%

Time: 10.1s

Alternatives: 14

Speedup: 2.4×

• 49.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in K around 0

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

   1. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \ell + \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.3)
     (fma (* (cos (* 0.5 K)) J) (+ l l) U)
     (if (<= t_0 -0.01)
       (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
       (fma (* (* (sinh l) 1.0) 2.0) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.3) {
		tmp = fma((cos((0.5 * K)) * J), (l + l), U);
	} else if (t_0 <= -0.01) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.3)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(l + l), U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.3], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 76.0%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 85.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

Alternative 3: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \ell + \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-4.340277777777778 \cdot 10^{-5}, K \cdot K, 0.005208333333333333\right), K \cdot K, -0.25 \cdot \ell\right), K \cdot K, 2 \cdot \ell\right) \cdot J}{U}, U, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.3)
     (fma (* (cos (* 0.5 K)) J) (+ l l) U)
     (if (<= t_0 -0.01)
       (fma
        (/
         (*
          (fma
           (fma
            (* l (fma -4.340277777777778e-5 (* K K) 0.005208333333333333))
            (* K K)
            (* -0.25 l))
           (* K K)
           (* 2.0 l))
          J)
         U)
        U
        U)
       (fma (* (* (sinh l) 1.0) 2.0) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.3) {
		tmp = fma((cos((0.5 * K)) * J), (l + l), U);
	} else if (t_0 <= -0.01) {
		tmp = fma(((fma(fma((l * fma(-4.340277777777778e-5, (K * K), 0.005208333333333333)), (K * K), (-0.25 * l)), (K * K), (2.0 * l)) * J) / U), U, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.3)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(l + l), U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(fma(fma(Float64(l * fma(-4.340277777777778e-5, Float64(K * K), 0.005208333333333333)), Float64(K * K), Float64(-0.25 * l)), Float64(K * K), Float64(2.0 * l)) * J) / U), U, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.3], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(N[(N[(l * N[(-4.340277777777778e-5 * N[(K * K), $MachinePrecision] + 0.005208333333333333), $MachinePrecision]), $MachinePrecision] * N[(K * K), $MachinePrecision] + N[(-0.25 * l), $MachinePrecision]), $MachinePrecision] * N[(K * K), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] / U), $MachinePrecision] * U + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 76.0%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 85.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 32.3

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(2 \cdot \left(J \cdot \ell\right) + {K}^{2} \cdot \left(\frac{-1}{4} \cdot \left(J \cdot \ell\right) + {K}^{2} \cdot \left(\frac{-1}{23040} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \frac{1}{192} \cdot \left(J \cdot \ell\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4.340277777777778 \cdot 10^{-5}, \left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, 0.005208333333333333 \cdot \left(J \cdot \ell\right)\right) \cdot K, K, -0.25 \cdot \left(J \cdot \ell\right)\right), \color{blue}{K \cdot K}, \mathsf{fma}\left(2 \cdot J, \ell, U\right)\right) \]

   9. Taylor expanded in U around inf

      \[\leadsto U \cdot \left(1 + \color{blue}{\left(2 \cdot \frac{J \cdot \ell}{U} + \frac{{K}^{2} \cdot \left(\frac{-1}{4} \cdot \left(J \cdot \ell\right) + {K}^{2} \cdot \left(\frac{-1}{23040} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \frac{1}{192} \cdot \left(J \cdot \ell\right)\right)\right)}{U}\right)}\right) \]

   11. Applied rewrites 77.7%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(-4.340277777777778 \cdot 10^{-5}, K \cdot K, 0.005208333333333333\right), K \cdot K, -0.25 \cdot \ell\right), K \cdot K, 2 \cdot \ell\right) \cdot J}{U}, U, U\right) \]

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

Alternative 4: 84.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.0%

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

   12. Applied rewrites 49.0%

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

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

Alternative 5: 80.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot J, K \cdot K, J\right), \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (fma (* -0.125 J) (* K K) J) (+ l l) U)
   (fma
    (*
     (*
      (*
       (fma
        (fma
         (fma 0.0001984126984126984 (* l l) 0.008333333333333333)
         (* l l)
         0.16666666666666666)
        (* l l)
        1.0)
       l)
      1.0)
     2.0)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(fma((-0.125 * J), (K * K), J), (l + l), U);
	} else {
		tmp = fma((((fma(fma(fma(0.0001984126984126984, (l * l), 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(fma(Float64(-0.125 * J), Float64(K * K), J), Float64(l + l), U);
	else
		tmp = fma(Float64(Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(l * l), 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(-0.125 * J), $MachinePrecision] * N[(K * K), $MachinePrecision] + J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(l * l), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.0%

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

   12. Applied rewrites 49.0%

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

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 93.3

           \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot 1\right) \cdot 2, J, U\right) \]

   13. Applied rewrites 93.3%

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

Alternative 6: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5e-5)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (+
    (*
     (*
      J
      (*
       (fma
        (fma
         (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5e-5) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5e-5)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5e-5], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 5.00000000000000024e-5

   1. Initial program 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.3

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

   7. Applied rewrites 78.3%

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

   if 5.00000000000000024e-5 < K

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

Alternative 7: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot J, K \cdot K, J\right), \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (fma (* -0.125 J) (* K K) J) (+ l l) U)
   (fma
    (*
     (*
      (*
       (fma (fma 0.008333333333333333 (* l l) 0.16666666666666666) (* l l) 1.0)
       l)
      1.0)
     2.0)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(fma((-0.125 * J), (K * K), J), (l + l), U);
	} else {
		tmp = fma((((fma(fma(0.008333333333333333, (l * l), 0.16666666666666666), (l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(fma(Float64(-0.125 * J), Float64(K * K), J), Float64(l + l), U);
	else
		tmp = fma(Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(-0.125 * J), $MachinePrecision] * N[(K * K), $MachinePrecision] + J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.0%

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

   12. Applied rewrites 49.0%

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

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 92.8

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

   13. Applied rewrites 92.8%

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

Alternative 8: 83.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5e-5)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (+
    (*
     (*
      J
      (*
       (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5e-5) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5e-5)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5e-5], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 5.00000000000000024e-5

   1. Initial program 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.3

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

   7. Applied rewrites 78.3%

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

   if 5.00000000000000024e-5 < K

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 9: 75.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot J, K \cdot K, J\right), \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (fma (* -0.125 J) (* K K) J) (+ l l) U)
   (fma (* (* (* (fma 0.16666666666666666 (* l l) 1.0) l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(fma((-0.125 * J), (K * K), J), (l + l), U);
	} else {
		tmp = fma((((fma(0.16666666666666666, (l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(fma(Float64(-0.125 * J), Float64(K * K), J), Float64(l + l), U);
	else
		tmp = fma(Float64(Float64(Float64(fma(0.16666666666666666, Float64(l * l), 1.0) * l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(-0.125 * J), $MachinePrecision] * N[(K * K), $MachinePrecision] + J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.0%

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

   12. Applied rewrites 49.0%

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

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

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 96.4%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 86.1

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

   13. Applied rewrites 86.1%

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

Alternative 10: 61.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.0%

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

   12. Applied rewrites 49.0%

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

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

   1. Initial program 85.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 61.8

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

   5. Applied rewrites 61.8%

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

   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 58.2%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 65.1%

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

Alternative 11: 87.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 8.4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 8.4e-14)
   (fma (* (* (sinh l) 1.0) 2.0) J U)
   (fma (* (* (cos (* 0.5 K)) (fma (* l l) 0.3333333333333333 2.0)) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 8.4e-14) {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	} else {
		tmp = fma(((cos((0.5 * K)) * fma((l * l), 0.3333333333333333, 2.0)) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 8.4e-14)
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(cos(Float64(0.5 * K)) * fma(Float64(l * l), 0.3333333333333333, 2.0)) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 8.4e-14], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 8.3999999999999995e-14

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 83.7%

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

   if 8.3999999999999995e-14 < K

   1. Initial program 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 93.5%

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

Alternative 12: 86.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 8.4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 8.4e-14)
   (fma (* (* (sinh l) 1.0) 2.0) J U)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 8.4e-14) {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	} else {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 8.4e-14)
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	else
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 8.4e-14], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 8.3999999999999995e-14

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 83.7%

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

   if 8.3999999999999995e-14 < K

   1. Initial program 83.7%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 93.6%

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

Alternative 13: 59.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5200000000 \lor \neg \left(\ell \leq 1.35 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot J, K \cdot K, J\right), \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5200000000.0) (not (<= l 1.35e-9)))
   (fma (fma (* -0.125 J) (* K K) J) (+ l l) U)
   (fma (+ J J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5200000000.0) || !(l <= 1.35e-9)) {
		tmp = fma(fma((-0.125 * J), (K * K), J), (l + l), U);
	} else {
		tmp = fma((J + J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5200000000.0) || !(l <= 1.35e-9))
		tmp = fma(fma(Float64(-0.125 * J), Float64(K * K), J), Float64(l + l), U);
	else
		tmp = fma(Float64(J + J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5200000000.0], N[Not[LessEqual[l, 1.35e-9]], $MachinePrecision]], N[(N[(N[(-0.125 * J), $MachinePrecision] * N[(K * K), $MachinePrecision] + J), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.2e9 or 1.3500000000000001e-9 < 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 25.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 34.6%

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

   12. Applied rewrites 34.6%

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

   if -5.2e9 < l < 1.3500000000000001e-9

   1. Initial program 65.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 98.9

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

   5. Applied rewrites 98.9%

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

   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 79.5%

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

   10. Applied rewrites 79.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5200000000 \lor \neg \left(\ell \leq 1.35 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot J, K \cdot K, J\right), \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.2% accurate, 33.0× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (fma (+ J J) l U))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. cos-neg-rev N/A

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

   14. lower-cos.f64 N/A

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

   15. distribute-lft-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 63.5

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

5. Applied rewrites 63.5%

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

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 50.0%

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

10. Applied rewrites 50.0%

    \[\leadsto \mathsf{fma}\left(J + J, \ell, U\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))