Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.9%

Time: 17.2s

Alternatives: 25

Speedup: 1.2×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

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

   7. lift-neg.f64 N/A

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

   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. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \ell \cdot \left(\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)\right)\\ \mathbf{if}\;\ell \leq -1.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (sinh l) (* (+ J J) (cos (* -0.5 K))))))
   (if (<= l -1.15)
     t_0
     (if (<= l 1.08e-7)
       (+
        (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0)))
        U)
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = sinh(l) * ((J + J) * cos((-0.5 * K)));
	double tmp;
	if (l <= -1.15) {
		tmp = t_0;
	} else if (l <= 1.08e-7) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(sinh(l) * Float64(Float64(J + J) * cos(Float64(-0.5 * K))))
	tmp = 0.0
	if (l <= -1.15)
		tmp = t_0;
	elseif (l <= 1.08e-7)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Sinh[l], $MachinePrecision] * N[(N[(J + J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.15], t$95$0, If[LessEqual[l, 1.08e-7], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1499999999999999 or 1.08000000000000001e-7 < l

   1. Initial program 99.9%

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

   2. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   3. 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. sinh-undef N/A

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

      3. count-2-rev N/A

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

      4. distribute-lft-in N/A

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

      5. distribute-rgt-out N/A

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

      6. count-2-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   if -1.1499999999999999 < l < 1.08000000000000001e-7

   1. Initial program 71.4%

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

   2. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 99.9%

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

Alternative 4: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \left(\left(t\_1 \cdot \ell\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+90}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 52:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot t\_1, J, U\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J J)) U))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* (* (* t_1 l) (fma (* l l) 0.3333333333333333 2.0)) J)))
   (if (<= l -4.4e+90)
     t_2
     (if (<= l -3.5e-5)
       t_0
       (if (<= l 52.0)
         (fma (* (+ l l) t_1) J U)
         (if (<= l 2.6e+89) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + J)), U);
	double t_1 = cos((0.5 * K));
	double t_2 = ((t_1 * l) * fma((l * l), 0.3333333333333333, 2.0)) * J;
	double tmp;
	if (l <= -4.4e+90) {
		tmp = t_2;
	} else if (l <= -3.5e-5) {
		tmp = t_0;
	} else if (l <= 52.0) {
		tmp = fma(((l + l) * t_1), J, U);
	} else if (l <= 2.6e+89) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + J)), U)
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(Float64(Float64(t_1 * l) * fma(Float64(l * l), 0.3333333333333333, 2.0)) * J)
	tmp = 0.0
	if (l <= -4.4e+90)
		tmp = t_2;
	elseif (l <= -3.5e-5)
		tmp = t_0;
	elseif (l <= 52.0)
		tmp = fma(Float64(Float64(l + l) * t_1), J, U);
	elseif (l <= 2.6e+89)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -4.4e+90], t$95$2, If[LessEqual[l, -3.5e-5], t$95$0, If[LessEqual[l, 52.0], N[(N[(N[(l + l), $MachinePrecision] * t$95$1), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 2.6e+89], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.39999999999999981e90 or 2.6000000000000001e89 < 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. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 97.4

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

   7. Applied rewrites 97.4%

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

   if -4.39999999999999981e90 < l < -3.4999999999999997e-5 or 52 < l < 2.6000000000000001e89

   1. Initial program 99.7%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 73.7%

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

   if -3.4999999999999997e-5 < l < 52

   1. Initial program 71.5%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 5: 87.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2060000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2060000.0)
   (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J 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 <= 2060000.0) {
		tmp = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + 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 <= 2060000.0)
		tmp = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + 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, 2060000.0], N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + 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 < 2.06e6

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 78.3%

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

   if 2.06e6 < K

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   3. 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 89.5

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

   4. Applied rewrites 89.5%

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

Alternative 6: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2060000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2060000.0)
   (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J J)) U)
   (fma (* (* (cos (* -0.5 K)) J) (fma (* l l) 0.3333333333333333 2.0)) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2060000.0) {
		tmp = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + J)), U);
	} else {
		tmp = fma(((cos((-0.5 * K)) * J) * fma((l * l), 0.3333333333333333, 2.0)), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2060000.0)
		tmp = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + J)), U);
	else
		tmp = fma(Float64(Float64(cos(Float64(-0.5 * K)) * J) * fma(Float64(l * l), 0.3333333333333333, 2.0)), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2060000.0], N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.06e6

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 78.3%

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

   if 2.06e6 < K

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 85.9%

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

Alternative 7: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot \cos \left(0.5 \cdot K\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \frac{\sinh \ell}{U}, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 (- INFINITY))
     (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J J)) U)
     (if (<= t_0 2e+150)
       (fma (* (+ l l) (cos (* 0.5 K))) J U)
       (* (fma (+ J J) (/ (sinh l) U) 1.0) 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 <= -((double) INFINITY)) {
		tmp = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + J)), U);
	} else if (t_0 <= 2e+150) {
		tmp = fma(((l + l) * cos((0.5 * K))), J, U);
	} else {
		tmp = fma((J + J), (sinh(l) / U), 1.0) * 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 <= Float64(-Inf))
		tmp = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + J)), U);
	elseif (t_0 <= 2e+150)
		tmp = fma(Float64(Float64(l + l) * cos(Float64(0.5 * K))), J, U);
	else
		tmp = Float64(fma(Float64(J + J), Float64(sinh(l) / U), 1.0) * 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, (-Infinity)], N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e+150], N[(N[(N[(l + l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 74.0%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1.99999999999999996e150

   1. Initial program 71.8%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 1.99999999999999996e150 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.6%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in U around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

   9. Applied rewrites 73.9%

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

Alternative 8: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \frac{\sinh \ell}{U}, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 (- INFINITY))
     (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J J)) U)
     (if (<= t_0 2e+150)
       (fma (* (+ l l) J) (cos (* -0.5 K)) U)
       (* (fma (+ J J) (/ (sinh l) U) 1.0) 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 <= -((double) INFINITY)) {
		tmp = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + J)), U);
	} else if (t_0 <= 2e+150) {
		tmp = fma(((l + l) * J), cos((-0.5 * K)), U);
	} else {
		tmp = fma((J + J), (sinh(l) / U), 1.0) * 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 <= Float64(-Inf))
		tmp = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + J)), U);
	elseif (t_0 <= 2e+150)
		tmp = fma(Float64(Float64(l + l) * J), cos(Float64(-0.5 * K)), U);
	else
		tmp = Float64(fma(Float64(J + J), Float64(sinh(l) / U), 1.0) * 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, (-Infinity)], N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e+150], N[(N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 74.0%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1.99999999999999996e150

   1. Initial program 71.8%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   if 1.99999999999999996e150 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.6%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in U around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

   9. Applied rewrites 73.9%

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

Alternative 9: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.66:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     U
     (if (<= t_0 -0.02)
       (fma
        (* (* (fma (* K K) -0.125 1.0) J) (fma (* l l) 0.3333333333333333 2.0))
        l
        U)
       (fma (sinh l) (+ J J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U;
	} else if (t_0 <= -0.02) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * J) * fma((l * l), 0.3333333333333333, 2.0)), l, U);
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = U;
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * J) * fma(Float64(l * l), 0.3333333333333333, 2.0)), l, U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], U, If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.660000000000000031

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{U} \]

   if -0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 58.5

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

   7. Applied rewrites 58.5%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 95.1%

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

Alternative 10: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.66:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot K, K, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     U
     (if (<= t_0 -0.02)
       (fma (* (fma (* -0.25 K) K 2.0) l) J U)
       (fma (sinh l) (+ J J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U;
	} else if (t_0 <= -0.02) {
		tmp = fma((fma((-0.25 * K), K, 2.0) * l), J, U);
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = U;
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(fma(Float64(-0.25 * K), K, 2.0) * l), J, U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], U, If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(-0.25 * K), $MachinePrecision] * K + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.660000000000000031

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{U} \]

   if -0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   12. Applied rewrites 56.4%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 95.1%

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

Alternative 11: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 64.3%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 95.1%

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

Alternative 12: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J + J\right) \cdot \sinh \ell\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (+ J J) (sinh l)))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 4e+22)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J + J) * sinh(l);
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 4e+22) {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J + J) * sinh(l))
	t_1 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 4e+22)
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 4e+22], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]]

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)))) < -inf.0 or 4e22 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.6%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. *-commutative N/A

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

      4. sinh-undef N/A

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

      5. count-2-rev N/A

         \[\leadsto J \cdot \left(\sinh \ell + \sinh \ell\right) \]

      6. distribute-lft-out N/A

         \[\leadsto J \cdot \sinh \ell + J \cdot \color{blue}{\sinh \ell} \]

      7. count-2-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

          \[\leadsto \left(J + J\right) \cdot \sinh \ell \]

      11. lift-+.f64 N/A

          \[\leadsto \left(J + J\right) \cdot \sinh \ell \]

      12. lift-sinh.f64 74.3

          \[\leadsto \left(J + J\right) \cdot \sinh \ell \]

   9. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 4e22

   1. Initial program 71.8%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 85.7

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

   7. Applied rewrites 85.7%

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

Alternative 13: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.66:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot K, K, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     U
     (if (<= t_0 -0.02)
       (fma (* (fma (* -0.25 K) K 2.0) l) J U)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U;
	} else if (t_0 <= -0.02) {
		tmp = fma((fma((-0.25 * K), K, 2.0) * l), J, U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = U;
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(fma(Float64(-0.25 * K), K, 2.0) * l), J, U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], U, If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(-0.25 * K), $MachinePrecision] * K + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.660000000000000031

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{U} \]

   if -0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   12. Applied rewrites 56.4%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 14: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.66:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     U
     (if (<= t_0 -0.02)
       (fma (* l J) (fma (* K K) -0.25 2.0) U)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U;
	} else if (t_0 <= -0.02) {
		tmp = fma((l * J), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = U;
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(l * J), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], U, If[LessEqual[t$95$0, -0.02], N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.660000000000000031

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{U} \]

   if -0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 52.8%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 15: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.66:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     U
     (if (<= t_0 -0.02)
       (* (* (* (* K K) J) -0.25) l)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U;
	} else if (t_0 <= -0.02) {
		tmp = (((K * K) * J) * -0.25) * l;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = U;
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(Float64(Float64(K * K) * J) * -0.25) * l);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], U, If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.660000000000000031

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{U} \]

   if -0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 52.8%

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

   11. Taylor expanded in K around inf

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       13. lift-*.f64 41.0

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]

   13. Applied rewrites 41.0%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]

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

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 16: 58.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(J + J\right) \cdot \ell\right) \cdot U}{U}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1280:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+234}:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (/ (* (* (+ J J) l) U) U)))
   (if (<= l -4.2e+59)
     t_0
     (if (<= l 1280.0)
       (fma (+ l l) J U)
       (if (<= l 4.9e+234) (* (* (* (* K K) J) -0.25) l) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (((J + J) * l) * U) / U;
	double tmp;
	if (l <= -4.2e+59) {
		tmp = t_0;
	} else if (l <= 1280.0) {
		tmp = fma((l + l), J, U);
	} else if (l <= 4.9e+234) {
		tmp = (((K * K) * J) * -0.25) * l;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(Float64(J + J) * l) * U) / U)
	tmp = 0.0
	if (l <= -4.2e+59)
		tmp = t_0;
	elseif (l <= 1280.0)
		tmp = fma(Float64(l + l), J, U);
	elseif (l <= 4.9e+234)
		tmp = Float64(Float64(Float64(Float64(K * K) * J) * -0.25) * l);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision]}, If[LessEqual[l, -4.2e+59], t$95$0, If[LessEqual[l, 1280.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 4.9e+234], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] * l), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.19999999999999968e59 or 4.89999999999999989e234 < 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. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 35.9

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 27.2

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

   10. Applied rewrites 27.2%

       \[\leadsto \left(J + J\right) \cdot \ell \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. count-2-rev N/A

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

       4. associate-*r* N/A

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

       5. *-rgt-identity N/A

          \[\leadsto \left(2 \cdot \left(J \cdot \ell\right)\right) \cdot 1 \]

       6. *-inverses N/A

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

       7. associate-/l* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. count-2-rev N/A

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

       12. lift-+.f64 N/A

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

       13. lift-*.f64 35.9

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

   12. Applied rewrites 35.9%

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

   if -4.19999999999999968e59 < l < 1280

   1. Initial program 74.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 91.9

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 1280 < l < 4.89999999999999989e234

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 23.7

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

   4. Applied rewrites 23.7%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 17.8

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

   7. Applied rewrites 17.8%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 31.1%

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

   11. Taylor expanded in K around inf

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       13. lift-*.f64 20.2

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]

   13. Applied rewrites 20.2%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* J (- (exp l) (exp (- l)))) (- INFINITY))
   (* (fma (* -0.25 K) K 2.0) (* l J))
   (* (fma (+ J J) (/ l U) 1.0) U)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 33.3%

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

   11. Taylor expanded in J around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 33.3

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

   13. Applied rewrites 33.3%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 81.3%

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

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

      7. lift-neg.f64 N/A

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

      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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 81.8%

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

   7. Taylor expanded in U around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)}{U}\right) \cdot U \]

   9. Applied rewrites 80.2%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 67.1%

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

Alternative 18: 57.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(\left(J + J\right) \cdot \ell\right) \cdot U}{U}\\ \mathbf{elif}\;\ell \leq 1280:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot K, K, 2\right) \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.2e+59)
   (/ (* (* (+ J J) l) U) U)
   (if (<= l 1280.0) (fma (+ l l) J U) (* (fma (* -0.25 K) K 2.0) (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e+59) {
		tmp = (((J + J) * l) * U) / U;
	} else if (l <= 1280.0) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((-0.25 * K), K, 2.0) * (l * J);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.2e+59)
		tmp = Float64(Float64(Float64(Float64(J + J) * l) * U) / U);
	elseif (l <= 1280.0)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = Float64(fma(Float64(-0.25 * K), K, 2.0) * Float64(l * J));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.2e+59], N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 1280.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(-0.25 * K), $MachinePrecision] * K + 2.0), $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.19999999999999968e59

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 32.7

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

   4. Applied rewrites 32.7%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 24.9

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 24.8

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

   10. Applied rewrites 24.8%

       \[\leadsto \left(J + J\right) \cdot \ell \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. count-2-rev N/A

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

       4. associate-*r* N/A

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

       5. *-rgt-identity N/A

          \[\leadsto \left(2 \cdot \left(J \cdot \ell\right)\right) \cdot 1 \]

       6. *-inverses N/A

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

       7. associate-/l* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. count-2-rev N/A

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

       12. lift-+.f64 N/A

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

       13. lift-*.f64 34.2

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

   12. Applied rewrites 34.2%

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

   if -4.19999999999999968e59 < l < 1280

   1. Initial program 74.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 91.9

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 1280 < 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. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 29.0

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

   4. Applied rewrites 29.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in J around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 34.2

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

   13. Applied rewrites 34.2%

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

Alternative 19: 56.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* J (- (exp l) (exp (- l)))) (- INFINITY))
   (* (fma (* -0.25 K) K 2.0) (* l J))
   (fma (/ (* (+ J J) l) U) U U)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 33.3%

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

   11. Taylor expanded in J around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 33.3

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

   13. Applied rewrites 33.3%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 81.3%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 75.3

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 63.8

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in U around inf

      \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right) \cdot U \]

      2. +-commutative N/A

         \[\leadsto \left(2 \cdot \frac{J \cdot \ell}{U} + 1\right) \cdot U \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(2 \cdot \frac{J \cdot \ell}{U}\right) \cdot U + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \frac{J \cdot \ell}{U}, U, U\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{2 \cdot \left(J \cdot \ell\right)}{U}, U, U\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2 \cdot \left(J \cdot \ell\right)}{U}, U, U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(2 \cdot J\right) \cdot \ell}{U}, U, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(2 \cdot J\right) \cdot \ell}{U}, U, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(J + J\right) \cdot \ell}{U}, U, U\right) \]

      10. lift-+.f64 65.1

          \[\leadsto \mathsf{fma}\left(\frac{\left(J + J\right) \cdot \ell}{U}, U, U\right) \]

   10. Applied rewrites 65.1%

       \[\leadsto \mathsf{fma}\left(\frac{\left(J + J\right) \cdot \ell}{U}, U, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 56.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J\\ \mathbf{elif}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(J + J\right) \cdot \ell}{U} \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (* (fma 2.0 l (/ U J)) J)
     (if (<= t_0 1e+304) (fma (+ l l) J U) (* (/ (* (+ J J) l) U) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(2.0, l, (U / J)) * J;
	} else if (t_0 <= 1e+304) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = (((J + J) * l) / U) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(2.0, l, Float64(U / J)) * J);
	elseif (t_0 <= 1e+304)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = Float64(Float64(Float64(Float64(J + J) * l) / U) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(2.0 * l + N[(U / J), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision] * U), $MachinePrecision]]]]

Derivation

1. Split input into 3 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) < -inf.0

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 28.5

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 21.7

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 21.7%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in J around inf

      \[\leadsto J \cdot \left(2 \cdot \ell + \color{blue}{\frac{U}{J}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

      4. lower-/.f64 27.1

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

   10. Applied rewrites 27.1%

       \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999994e303

   1. Initial program 71.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 99.0

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 85.1

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 85.1%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   if 9.9999999999999994e303 < (+.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. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 29.1

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 22.1

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 22.1%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in J around inf

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \ell \]

      3. count-2-rev N/A

         \[\leadsto \left(J + J\right) \cdot \ell \]

      4. lift-+.f64 21.6

         \[\leadsto \left(J + J\right) \cdot \ell \]

   10. Applied rewrites 21.6%

       \[\leadsto \left(J + J\right) \cdot \ell \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(J + J\right) \cdot \ell \]

       2. lift-+.f64 N/A

          \[\leadsto \left(J + J\right) \cdot \ell \]

       3. count-2-rev N/A

          \[\leadsto \left(2 \cdot J\right) \cdot \ell \]

       4. associate-*r* N/A

          \[\leadsto 2 \cdot \left(J \cdot \ell\right) \]

       5. *-rgt-identity N/A

          \[\leadsto \left(2 \cdot \left(J \cdot \ell\right)\right) \cdot 1 \]

       6. *-inverses N/A

          \[\leadsto \left(2 \cdot \left(J \cdot \ell\right)\right) \cdot \frac{U}{U} \]

       7. associate-/l* N/A

          \[\leadsto \frac{\left(2 \cdot \left(J \cdot \ell\right)\right) \cdot U}{U} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \left(J \cdot \ell\right)}{U} \cdot U \]

       9. associate-*r/ N/A

          \[\leadsto \left(2 \cdot \frac{J \cdot \ell}{U}\right) \cdot U \]

       10. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \frac{J \cdot \ell}{U}\right) \cdot U \]

       11. associate-*r/ N/A

           \[\leadsto \frac{2 \cdot \left(J \cdot \ell\right)}{U} \cdot U \]

       12. lower-/.f64 N/A

           \[\leadsto \frac{2 \cdot \left(J \cdot \ell\right)}{U} \cdot U \]

       13. associate-*r* N/A

           \[\leadsto \frac{\left(2 \cdot J\right) \cdot \ell}{U} \cdot U \]

       14. count-2-rev N/A

           \[\leadsto \frac{\left(J + J\right) \cdot \ell}{U} \cdot U \]

       15. lift-+.f64 N/A

           \[\leadsto \frac{\left(J + J\right) \cdot \ell}{U} \cdot U \]

       16. lift-*.f64 30.3

           \[\leadsto \frac{\left(J + J\right) \cdot \ell}{U} \cdot U \]

   12. Applied rewrites 30.3%

       \[\leadsto \frac{\left(J + J\right) \cdot \ell}{U} \cdot U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 54.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1280:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1280.0)
   (fma (+ l l) J U)
   (if (<= l 1.5e+233)
     (* (* (* (* K K) J) -0.25) l)
     (* (fma 2.0 l (/ U J)) J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1280.0) {
		tmp = fma((l + l), J, U);
	} else if (l <= 1.5e+233) {
		tmp = (((K * K) * J) * -0.25) * l;
	} else {
		tmp = fma(2.0, l, (U / J)) * J;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1280.0)
		tmp = fma(Float64(l + l), J, U);
	elseif (l <= 1.5e+233)
		tmp = Float64(Float64(Float64(Float64(K * K) * J) * -0.25) * l);
	else
		tmp = Float64(fma(2.0, l, Float64(U / J)) * J);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1280.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 1.5e+233], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 * l + N[(U / J), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1280

   1. Initial program 81.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 76.1

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 64.5

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   if 1280 < l < 1.50000000000000007e233

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 23.6

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 17.7

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 17.7%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]

   10. Applied rewrites 31.0%

       \[\leadsto \mathsf{fma}\left(\ell \cdot J, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.25, 2\right)}, U\right) \]

   11. Taylor expanded in K around inf

       \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \left(\left(J \cdot {K}^{2}\right) \cdot \ell\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \ell \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot J\right)\right) \cdot \ell \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot J\right) \cdot \ell \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot J\right) \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)\right) \cdot \ell \]

       7. *-commutative N/A

          \[\leadsto \left(J \cdot \left({K}^{2} \cdot \frac{-1}{4}\right)\right) \cdot \ell \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       10. *-commutative N/A

           \[\leadsto \left(\left({K}^{2} \cdot J\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left({K}^{2} \cdot J\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       12. pow2 N/A

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot \frac{-1}{4}\right) \cdot \ell \]

       13. lift-*.f64 20.2

           \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]

   13. Applied rewrites 20.2%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.25\right) \cdot \ell \]

   if 1.50000000000000007e233 < 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. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 46.1

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 35.1

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in J around inf

      \[\leadsto J \cdot \left(2 \cdot \ell + \color{blue}{\frac{U}{J}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

      4. lower-/.f64 40.1

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

   10. Applied rewrites 40.1%

       \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) (- INFINITY))
   (* (fma 2.0 l (/ U J)) J)
   (fma (+ l l) J 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) <= -((double) INFINITY)) {
		tmp = fma(2.0, l, (U / J)) * J;
	} else {
		tmp = fma((l + l), J, 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) <= Float64(-Inf))
		tmp = Float64(fma(2.0, l, Float64(U / J)) * J);
	else
		tmp = fma(Float64(l + l), J, 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], (-Infinity)], N[(N[(2.0 * l + N[(U / J), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * J + 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) < -inf.0

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 28.5

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 21.7

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 21.7%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in J around inf

      \[\leadsto J \cdot \left(2 \cdot \ell + \color{blue}{\frac{U}{J}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

      4. lower-/.f64 27.1

         \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

   10. Applied rewrites 27.1%

       \[\leadsto \mathsf{fma}\left(2, \ell, \frac{U}{J}\right) \cdot J \]

   if -inf.0 < (+.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 81.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 75.7

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 53.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell + \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (+ l l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((l + l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(l + l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
3. 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. associate-*r* N/A

      \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

   9. count-2-rev N/A

      \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

   11. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

   15. metadata-eval 63.9

       \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

4. Applied rewrites 63.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

5. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

   2. count-2-rev N/A

      \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

   3. distribute-lft-out N/A

      \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

   4. count-2-rev N/A

      \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

   5. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

   7. count-2-rev N/A

      \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   8. lift-+.f64 53.5

      \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

7. Applied rewrites 53.5%

   \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]
8. Add Preprocessing

Alternative 24: 44.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J + J\right) \cdot \ell\\ \mathbf{if}\;\ell \leq -1600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-38}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (+ J J) l))) (if (<= l -1600.0) t_0 (if (<= l 7e-38) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J + J) * l;
	double tmp;
	if (l <= -1600.0) {
		tmp = t_0;
	} else if (l <= 7e-38) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	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 + j) * l
    if (l <= (-1600.0d0)) then
        tmp = t_0
    else if (l <= 7d-38) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (J + J) * l;
	double tmp;
	if (l <= -1600.0) {
		tmp = t_0;
	} else if (l <= 7e-38) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (J + J) * l
	tmp = 0
	if l <= -1600.0:
		tmp = t_0
	elif l <= 7e-38:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J + J) * l)
	tmp = 0.0
	if (l <= -1600.0)
		tmp = t_0;
	elseif (l <= 7e-38)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (J + J) * l;
	tmp = 0.0;
	if (l <= -1600.0)
		tmp = t_0;
	elseif (l <= 7e-38)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[l, -1600.0], t$95$0, If[LessEqual[l, 7e-38], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1600 or 7.0000000000000003e-38 < l

   1. Initial program 98.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   3. 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. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(\ell \cdot J\right), \cos \left(\frac{1}{2} \cdot \color{blue}{K}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}, U\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\color{blue}{\frac{1}{2}} \cdot K\right), U\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right), U\right) \]

      15. metadata-eval 32.5

          \[\leadsto \mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right) \]

   4. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \ell\right) + U \]

      2. count-2-rev N/A

         \[\leadsto \left(J \cdot \ell + J \cdot \ell\right) + U \]

      3. distribute-lft-out N/A

         \[\leadsto J \cdot \left(\ell + \ell\right) + U \]

      4. count-2-rev N/A

         \[\leadsto J \cdot \left(2 \cdot \ell\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot J + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]

      7. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

      8. lift-+.f64 24.9

         \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   7. Applied rewrites 24.9%

      \[\leadsto \mathsf{fma}\left(\ell + \ell, \color{blue}{J}, U\right) \]

   8. Taylor expanded in J around inf

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot J\right) \cdot \ell \]

      3. count-2-rev N/A

         \[\leadsto \left(J + J\right) \cdot \ell \]

      4. lift-+.f64 21.8

         \[\leadsto \left(J + J\right) \cdot \ell \]

   10. Applied rewrites 21.8%

       \[\leadsto \left(J + J\right) \cdot \ell \]

   if -1600 < l < 7.0000000000000003e-38

   1. Initial program 72.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   3. Step-by-step derivation

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 36.1% accurate, 68.7× 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 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
3. Step-by-step derivation

4. Applied rewrites 36.1%

   \[\leadsto \color{blue}{U} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))