Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.5%

Time: 6.7s

Alternatives: 24

Speedup: 2.4×

• 49.1% 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: 85.9% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\left(\left(\sinh \ell \cdot 2\right) \cdot t\_1\right) \cdot J\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot t\_1, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))))
        (t_1 (cos (* 0.5 K))))
   (if (<= t_0 -1e-29)
     (* (* (* (sinh l) 2.0) t_1) J)
     (if (<= t_0 2e+55)
       (fma (* (* l J) t_1) 2.0 U)
       (* (* t_1 J) (* 2.0 (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double t_1 = cos((0.5 * K));
	double tmp;
	if (t_0 <= -1e-29) {
		tmp = ((sinh(l) * 2.0) * t_1) * J;
	} else if (t_0 <= 2e+55) {
		tmp = fma(((l * J) * t_1), 2.0, U);
	} else {
		tmp = (t_1 * J) * (2.0 * sinh(l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	t_1 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_0 <= -1e-29)
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * t_1) * J);
	elseif (t_0 <= 2e+55)
		tmp = fma(Float64(Float64(l * J) * t_1), 2.0, U);
	else
		tmp = Float64(Float64(t_1 * J) * Float64(2.0 * sinh(l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-29], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[t$95$0, 2e+55], N[(N[(N[(l * J), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(t$95$1 * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -9.99999999999999943e-30

   1. Initial program 99.2%

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

      1. 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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

         \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)\right) \]

      2. sinh-undef-rev N/A

         \[\leadsto J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)\right) \]

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \left(\left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      7. rec-exp N/A

         \[\leadsto \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      8. sinh-undef-rev N/A

         \[\leadsto \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sinh \ell \cdot 2\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\sinh \ell \cdot 2\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      11. lift-sinh.f64 N/A

          \[\leadsto \left(\left(\sinh \ell \cdot 2\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(\sinh \ell \cdot 2\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J \]

      13. lift-*.f64 100.0

          \[\leadsto \left(\left(\sinh \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J \]

   7. Applied rewrites 100.0%

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

   if -9.99999999999999943e-30 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.00000000000000002e55

   1. Initial program 68.3%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 2.00000000000000002e55 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(\color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(\color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(e^{\color{blue}{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

         \[\leadsto \left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right) \]

   5. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-29} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot t\_1, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))))
        (t_1 (cos (* 0.5 K))))
   (if (or (<= t_0 -1e-29) (not (<= t_0 2e+55)))
     (* (* t_1 J) (* 2.0 (sinh l)))
     (fma (* (* l J) t_1) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double t_1 = cos((0.5 * K));
	double tmp;
	if ((t_0 <= -1e-29) || !(t_0 <= 2e+55)) {
		tmp = (t_1 * J) * (2.0 * sinh(l));
	} else {
		tmp = fma(((l * J) * t_1), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	t_1 = cos(Float64(0.5 * K))
	tmp = 0.0
	if ((t_0 <= -1e-29) || !(t_0 <= 2e+55))
		tmp = Float64(Float64(t_1 * J) * Float64(2.0 * sinh(l)));
	else
		tmp = fma(Float64(Float64(l * J) * t_1), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-29], N[Not[LessEqual[t$95$0, 2e+55]], $MachinePrecision]], N[(N[(t$95$1 * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * J), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -9.99999999999999943e-30 or 2.00000000000000002e55 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(\color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(\color{blue}{e^{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(e^{\color{blue}{\ell}} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \]

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

         \[\leadsto \left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right) \]

   5. Applied rewrites 100.0%

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

   if -9.99999999999999943e-30 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.00000000000000002e55

   1. Initial program 68.3%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq -1 \cdot 10^{-29} \lor \neg \left(\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 -2e+121)
     (fma
      J
      (*
       (*
        (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
        l)
       (fma (* K K) -0.125 1.0))
      U)
     (if (<= t_0 1e+15)
       (fma (* (* l J) (cos (* 0.5 K))) 2.0 U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -2e+121) {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), U);
	} else if (t_0 <= 1e+15) {
		tmp = fma(((l * J) * cos((0.5 * K))), 2.0, U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= -2e+121)
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	elseif (t_0 <= 1e+15)
		tmp = fma(Float64(Float64(l * J) * cos(Float64(0.5 * K))), 2.0, U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+121], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 1e+15], N[(N[(N[(l * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -2.00000000000000007e121

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 69.2

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

   10. Applied rewrites 69.2%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 72.5

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

   13. Applied rewrites 72.5%

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

   if -2.00000000000000007e121 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1e15

   1. Initial program 68.7%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 1e15 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 -2e+121)
     (fma
      J
      (*
       (*
        (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
        l)
       (fma (* K K) -0.125 1.0))
      U)
     (if (<= t_0 1e+15)
       (fma (* l (* (cos (* 0.5 K)) J)) 2.0 U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -2e+121) {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), U);
	} else if (t_0 <= 1e+15) {
		tmp = fma((l * (cos((0.5 * K)) * J)), 2.0, U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= -2e+121)
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	elseif (t_0 <= 1e+15)
		tmp = fma(Float64(l * Float64(cos(Float64(0.5 * K)) * J)), 2.0, U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+121], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 1e+15], N[(N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -2.00000000000000007e121

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 69.2

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

   10. Applied rewrites 69.2%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 72.5

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

   13. Applied rewrites 72.5%

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

   if -2.00000000000000007e121 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1e15

   1. Initial program 68.7%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 1e15 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-7} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (or (<= t_0 -2e-7) (not (<= t_0 5e+55))) (* (* l J) 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 5e+55)) {
		tmp = (l * J) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    if ((t_0 <= (-2d-7)) .or. (.not. (t_0 <= 5d+55))) then
        tmp = (l * j) * 2.0d0
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 5e+55)) {
		tmp = (l * J) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if (t_0 <= -2e-7) or not (t_0 <= 5e+55):
		tmp = (l * J) * 2.0
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if ((t_0 <= -2e-7) || !(t_0 <= 5e+55))
		tmp = Float64(Float64(l * J) * 2.0);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if ((t_0 <= -2e-7) || ~((t_0 <= 5e+55)))
		tmp = (l * J) * 2.0;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-7], N[Not[LessEqual[t$95$0, 5e+55]], $MachinePrecision]], N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.9999999999999999e-7 or 5.00000000000000046e55 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 25.1

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

   5. Applied rewrites 25.1%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 25.3

         \[\leadsto \left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 2 \]

   8. Applied rewrites 25.3%

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

   9. Taylor expanded in K around 0

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

       1. *-commutative N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot 2 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot 2 \]

       3. *-commutative N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]

       4. lift-*.f64 17.4

          \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]

   11. Applied rewrites 17.4%

       \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]

   if -1.9999999999999999e-7 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 5.00000000000000046e55

   1. Initial program 68.7%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -2 \cdot 10^{-7} \lor \neg \left(J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 5 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 2 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 2e+278)
   (fma (* l J) 2.0 U)
   (fma (* l (* J (fma (* K K) -0.125 1.0))) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 2e+278) {
		tmp = fma((l * J), 2.0, U);
	} else {
		tmp = fma((l * (J * fma((K * K), -0.125, 1.0))), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 2e+278)
		tmp = fma(Float64(l * J), 2.0, U);
	else
		tmp = fma(Float64(l * Float64(J * fma(Float64(K * K), -0.125, 1.0))), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[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], 2e+278], N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(l * N[(J * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.99999999999999993e278

   1. Initial program 76.5%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if 1.99999999999999993e278 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   10. Applied rewrites 36.6%

       \[\leadsto \mathsf{fma}\left(\ell \cdot \left(J \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right), 2, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) 5e-270)
   (fma (* l J) 2.0 U)
   (fma J (* (* 2.0 l) (fma (* K K) -0.125 1.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (((J * (exp(l) - exp(-l))) * cos((K / 2.0))) <= 5e-270) {
		tmp = fma((l * J), 2.0, U);
	} else {
		tmp = fma(J, ((2.0 * l) * fma((K * K), -0.125, 1.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) <= 5e-270)
		tmp = fma(Float64(l * J), 2.0, U);
	else
		tmp = fma(J, Float64(Float64(2.0 * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-270], N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(N[(2.0 * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 4.9999999999999998e-270

   1. Initial program 76.2%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 4.9999999999999998e-270 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 39.9%

       \[\leadsto \mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma
    J
    (*
     (*
      (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
      l)
     (fma (* K K) -0.125 1.0))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 51.3

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

   13. Applied rewrites 51.3%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

   1. 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. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 10: 89.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right) \cdot \ell, \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) J U)
   (+
    (*
     (*
      (*
       (fma
        (*
         (fma
          (fma (* l l) 0.0003968253968253968 0.016666666666666666)
          (* l l)
          0.3333333333333333)
         l)
        l
        2.0)
       J)
      l)
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = (((fma((fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333) * l), l, 2.0) * J) * l) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333) * l), l, 2.0) * J) * l) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * l + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.4999999999999999e-14

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.6%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 95.6

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

   10. Applied rewrites 95.6%

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

Alternative 11: 83.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot 1 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma
    J
    (*
     (*
      (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
      l)
     (fma (* K K) -0.125 1.0))
    U)
   (+
    (*
     (*
      (*
       (fma
        (fma
         (fma (* l l) 0.0003968253968253968 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       J)
      l)
     1.0)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = (((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * J) * l) * 1.0) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * J) * l) * 1.0) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 51.3

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

   13. Applied rewrites 51.3%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 94.2%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 89.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \color{blue}{1} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot 1 + U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma
    J
    (*
     (*
      (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
      l)
     (fma (* K K) -0.125 1.0))
    U)
   (fma
    J
    (*
     (fma
      (fma
       (fma (* l l) 0.0003968253968253968 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, (fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 51.3

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

   13. Applied rewrites 51.3%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 89.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) J U)
   (+
    (*
     (*
      (*
       (fma
        (fma
         (fma (* l l) 0.0003968253968253968 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       J)
      l)
     (cos (* 0.5 K)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = (((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * J) * l) * cos((0.5 * K))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * J) * l) * cos(Float64(0.5 * K))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.4999999999999999e-14

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in K around 0

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

       1. lift-*.f64 95.6

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

   11. Applied rewrites 95.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma J (* (* (* (* l l) 0.3333333333333333) l) (fma (* K K) -0.125 1.0)) U)
   (fma
    J
    (*
     (fma
      (fma
       (fma (* l l) 0.0003968253968253968 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, (fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.5

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

   13. Applied rewrites 50.5%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 89.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) 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 <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), 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 <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), 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, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $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 < 1.4999999999999999e-14

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 16: 88.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) J U)
   (fma
    J
    (*
     (*
      (fma (fma (* l l) 0.016666666666666666 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 <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = fma(J, ((fma(fma((l * l), 0.016666666666666666, 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 <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = fma(J, Float64(Float64(fma(fma(Float64(l * l), 0.016666666666666666, 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, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.4999999999999999e-14

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 95.3

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

   7. Applied rewrites 95.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma J (* (* (* (* l l) 0.3333333333333333) l) (fma (* K K) -0.125 1.0)) U)
   (fma
    J
    (*
     (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, (fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.5

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

   13. Applied rewrites 50.5%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 87.9

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

   10. Applied rewrites 87.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma J (* (* 2.0 l) (fma (* K K) -0.125 1.0)) U)
   (fma
    J
    (*
     (fma (fma (* l l) 0.016666666666666666 0.3333333333333333) (* l l) 2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((2.0 * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, (fma(fma((l * l), 0.016666666666666666, 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(2.0 * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(fma(fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(2.0 * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 47.0%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 87.9

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

   10. Applied rewrites 87.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 87.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) J U)
   (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 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 < 1.4999999999999999e-14

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 20: 77.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma J (* (* 2.0 l) (fma (* K K) -0.125 1.0)) U)
   (fma J (* (* (fma (* l l) 0.3333333333333333 2.0) l) 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((2.0 * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, ((fma((l * l), 0.3333333333333333, 2.0) * l) * 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(2.0 * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(2.0 * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 47.0%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      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. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lift-exp.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. lift-/.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)} + U \]

      9. lift-cos.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)} + U \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left({\ell}^{2} \cdot \frac{1}{3} + 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-*.f64 91.8

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   7. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \color{blue}{1}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \color{blue}{1}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (* (* l J) (* (* K K) -0.125)) 2.0 U)
   (fma (* l J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(((l * J) * ((K * K) * -0.125)), 2.0, U);
	} else {
		tmp = fma((l * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(Float64(l * J) * Float64(Float64(K * K) * -0.125)), 2.0, U);
	else
		tmp = fma(Float64(l * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 76.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot 2 + U \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \color{blue}{2}, U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      9. lower-*.f64 66.3

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right) \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right), 2, U\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\frac{-1}{8} \cdot {K}^{2} + 1\right), 2, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right), 2, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right), 2, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), 2, U\right) \]

      5. lower-*.f64 44.0

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right) \]

   8. Applied rewrites 44.0%

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right) \]

   9. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\frac{-1}{8} \cdot {K}^{2}\right), 2, U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left({K}^{2} \cdot \frac{-1}{8}\right), 2, U\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left({K}^{2} \cdot \frac{-1}{8}\right), 2, U\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot \frac{-1}{8}\right), 2, U\right) \]

       4. lift-*.f64 44.0

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), 2, U\right) \]

   11. Applied rewrites 44.0%

       \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125\right), 2, U\right) \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot 2 + U \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \color{blue}{2}, U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

      9. lower-*.f64 65.9

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right) \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(J \cdot \ell, 2, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]

      2. lift-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]

   8. Applied rewrites 60.9%

      \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 87.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.5e-14)
   (fma (* 2.0 (sinh l)) J U)
   (fma J (* (* (fma (* l l) 0.3333333333333333 2.0) l) (cos (* 0.5 K))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.5e-14) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = fma(J, ((fma((l * l), 0.3333333333333333, 2.0) * l) * cos((0.5 * K))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1.5e-14)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = fma(J, Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * cos(Float64(0.5 * K))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1.5e-14], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.4999999999999999e-14

   1. Initial program 85.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot J + U \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}, \color{blue}{J}, U\right) \]

      4. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right) \]

      6. lower-sinh.f64 84.9

         \[\leadsto \mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)} \]

   if 1.4999999999999999e-14 < K

   1. Initial program 74.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lift--.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lift-exp.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - \color{blue}{e^{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. lift-/.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)} + U \]

      9. lift-cos.f64 N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)} + U \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left({\ell}^{2} \cdot \frac{1}{3} + 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-*.f64 95.0

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   7. Applied rewrites 95.0%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 95.0

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot \color{blue}{K}\right), U\right) \]

   10. Applied rewrites 95.0%

       \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 54.3% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell \cdot J, 2, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (* l J) 2.0 U))

C

double code(double J, double l, double K, double U) {
	return fma((l * J), 2.0, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(l * J), 2.0, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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

      \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + \color{blue}{U} \]

   2. *-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot 2 + U \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \color{blue}{2}, U\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

   8. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), 2, U\right) \]

   9. lower-*.f64 66.0

      \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right) \]

5. Applied rewrites 66.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto \mathsf{fma}\left(J \cdot \ell, 2, U\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]

   2. lift-*.f64 52.3

      \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]

8. Applied rewrites 52.3%

   \[\leadsto \mathsf{fma}\left(\ell \cdot J, 2, U\right) \]
9. Add Preprocessing

Alternative 24: 36.7% accurate, 330.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

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 = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 82.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))