Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.9%

Time: 8.8s

Alternatives: 20

Speedup: 2.1×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 87.8

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sinh-undef N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sinh.f64 100.0

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

   14. lift-cos.f64 N/A

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

   15. cos-neg-rev N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f64 N/A

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

   20. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in K around inf

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.7138:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.7138)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.7138) {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.7138)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.7138], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.71379999999999999

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 92.8%

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

   if 0.71379999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 89.0

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.5

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

   9. Applied rewrites 99.5%

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

Alternative 3: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.7138:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.7138)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.7138) {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.7138)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.7138], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.71379999999999999

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 91.8%

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

   if 0.71379999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 89.0

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.5

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

   9. Applied rewrites 99.5%

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

Alternative 4: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.191:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.191)
     (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_0) U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.191) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.191)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.191], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.191

   1. Initial program 84.7%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 88.2%

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

   if 0.191 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 88.8

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 5: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.191:\\ \;\;\;\;\left(\left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right) \cdot \ell\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.191)
     (+ (* (* (* J (fma (* l l) 0.3333333333333333 2.0)) l) t_0) U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.191) {
		tmp = (((J * fma((l * l), 0.3333333333333333, 2.0)) * l) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.191)
		tmp = Float64(Float64(Float64(Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0)) * l) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.191], N[(N[(N[(N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.191

   1. Initial program 84.7%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 86.7%

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

   if 0.191 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 88.8

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 6: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.191:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.191)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.191) {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.191)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.191], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.191

   1. Initial program 84.7%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 86.7%

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

   if 0.191 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 88.8

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 7: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma (* (+ J J) l) (cos (* 0.5 K)) U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma(((J + J) * l), cos((0.5 * K)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 85.3%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 85.3

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 99.9

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around inf

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 71.6%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 88.5

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.1

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

   9. Applied rewrites 97.1%

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

Alternative 8: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot J\right) \cdot \ell\right) \cdot 0.3333333333333333\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l J) l) 0.3333333333333333)) l U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * J) * l) * 0.3333333333333333)), l, U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * J) * l) * 0.3333333333333333)), l, U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * J), $MachinePrecision] * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 61.1%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 88.1

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.7

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

   9. Applied rewrites 96.7%

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

Alternative 9: 80.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot J\right) \cdot \ell\right) \cdot 0.3333333333333333\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell}{U} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l J) l) 0.3333333333333333)) l U)
   (* U (- (* J (/ (* (fma (* l l) 0.3333333333333333 2.0) l) U)) -1.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * J) * l) * 0.3333333333333333)), l, U);
	} else {
		tmp = U * ((J * ((fma((l * l), 0.3333333333333333, 2.0) * l) / U)) - -1.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * J) * l) * 0.3333333333333333)), l, U);
	else
		tmp = Float64(U * Float64(Float64(J * Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) / U)) - -1.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * J), $MachinePrecision] * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(U * N[(N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 61.1%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 89.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot J\right) \cdot \ell\right) \cdot 0.3333333333333333\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell}{U} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ t_2 := \left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, t\_0, U\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (fma (* (+ J J) (sinh l)) 1.0 U))
        (t_2 (* (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0)))
   (if (<= l -6e+99)
     t_2
     (if (<= l -50.0)
       t_1
       (if (<= l 2.9e-5)
         (fma (* (+ J J) l) t_0 U)
         (if (<= l 8.2e+102) t_1 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = fma(((J + J) * sinh(l)), 1.0, U);
	double t_2 = ((fma((l * l), 0.3333333333333333, 2.0) * l) * J) * t_0;
	double tmp;
	if (l <= -6e+99) {
		tmp = t_2;
	} else if (l <= -50.0) {
		tmp = t_1;
	} else if (l <= 2.9e-5) {
		tmp = fma(((J + J) * l), t_0, U);
	} else if (l <= 8.2e+102) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U)
	t_2 = Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) * t_0)
	tmp = 0.0
	if (l <= -6e+99)
		tmp = t_2;
	elseif (l <= -50.0)
		tmp = t_1;
	elseif (l <= 2.9e-5)
		tmp = fma(Float64(Float64(J + J) * l), t_0, U);
	elseif (l <= 8.2e+102)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[l, -6e+99], t$95$2, If[LessEqual[l, -50.0], t$95$1, If[LessEqual[l, 2.9e-5], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * t$95$0 + U), $MachinePrecision], If[LessEqual[l, 8.2e+102], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.00000000000000029e99 or 8.1999999999999999e102 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in J around inf

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

   8. Applied rewrites 98.9%

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

   if -6.00000000000000029e99 < l < -50 or 2.9e-5 < l < 8.1999999999999999e102

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if -50 < l < 2.9e-5

   1. Initial program 75.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 75.1

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sinh.f64 100.0

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around inf

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 99.9%

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

Alternative 11: 79.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot J\right) \cdot \ell\right) \cdot 0.3333333333333333\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J}{U}, U, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l J) l) 0.3333333333333333)) l U)
   (fma (/ (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) U) U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * J) * l) * 0.3333333333333333)), l, U);
	} else {
		tmp = fma((((fma((l * l), 0.3333333333333333, 2.0) * l) * J) / U), U, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * J) * l) * 0.3333333333333333)), l, U);
	else
		tmp = fma(Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) / U), U, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * J), $MachinePrecision] * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] / U), $MachinePrecision] * U + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 61.1%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 86.7%

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

Alternative 12: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot J\right) \cdot \ell\right) \cdot 0.3333333333333333\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l J) l) 0.3333333333333333)) l U)
   (fma (* (fma (* 0.3333333333333333 l) l 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * J) * l) * 0.3333333333333333)), l, U);
	} else {
		tmp = fma((fma((0.3333333333333333 * l), l, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * J) * l) * 0.3333333333333333)), l, U);
	else
		tmp = fma(Float64(fma(Float64(0.3333333333333333 * l), l, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * J), $MachinePrecision] * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 61.1%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.3%

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

Alternative 13: 73.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\left(J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.82)
   (* (* J (* (* K K) -0.125)) (* (fma (* l l) 0.3333333333333333 2.0) l))
   (fma (* (fma (* 0.3333333333333333 l) l 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.82) {
		tmp = (J * ((K * K) * -0.125)) * (fma((l * l), 0.3333333333333333, 2.0) * l);
	} else {
		tmp = fma((fma((0.3333333333333333 * l), l, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.82)
		tmp = Float64(Float64(J * Float64(Float64(K * K) * -0.125)) * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l));
	else
		tmp = fma(Float64(fma(Float64(0.3333333333333333 * l), l, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.82], N[(N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 13.8%

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

   9. Taylor expanded in K around inf

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

   11. Applied rewrites 43.2%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 79.2%

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

Alternative 14: 73.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.82)
   (fma (* (fma (* K K) -0.125 1.0) (* J 2.0)) l U)
   (fma (* (fma (* 0.3333333333333333 l) l 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.82) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (J * 2.0)), l, U);
	} else {
		tmp = fma((fma((0.3333333333333333 * l), l, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.82)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(J * 2.0)), l, U);
	else
		tmp = fma(Float64(fma(Float64(0.3333333333333333 * l), l, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.82], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 43.2%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 79.2%

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

Alternative 15: 72.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -145000.0) (not (<= l 8e+21)))
   (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
   (fma (* 2.0 l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -145000.0) || !(l <= 8e+21)) {
		tmp = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	} else {
		tmp = fma((2.0 * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -145000.0) || !(l <= 8e+21))
		tmp = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J);
	else
		tmp = fma(Float64(2.0 * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -145000.0], N[Not[LessEqual[l, 8e+21]], $MachinePrecision]], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -145000 or 8e21 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 65.4%

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

   if -145000 < l < 8e21

   1. Initial program 76.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 83.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.3333333333333333 \cdot \ell\right) \cdot \ell\right) \cdot \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.6e-7)
   (fma (* (* (* 0.3333333333333333 l) l) l) J U)
   (if (<= l 8e+21)
     (fma (* 2.0 l) J U)
     (* (* (fma (* l l) 0.3333333333333333 2.0) l) J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.6e-7) {
		tmp = fma((((0.3333333333333333 * l) * l) * l), J, U);
	} else if (l <= 8e+21) {
		tmp = fma((2.0 * l), J, U);
	} else {
		tmp = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.6e-7)
		tmp = fma(Float64(Float64(Float64(0.3333333333333333 * l) * l) * l), J, U);
	elseif (l <= 8e+21)
		tmp = fma(Float64(2.0 * l), J, U);
	else
		tmp = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.6e-7], N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 8e+21], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.6e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 68.3%

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

   12. Taylor expanded in l around inf

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

   14. Applied rewrites 68.3%

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

   if -1.6e-7 < l < 8e21

   1. Initial program 75.7%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 86.2%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 86.2%

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

   if 8e21 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 57.7%

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

Alternative 17: 70.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;\ell \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -145000.0) (not (<= l 8e+21)))
   (* l (* (* (* l l) 0.3333333333333333) J))
   (fma (* 2.0 l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -145000.0) || !(l <= 8e+21)) {
		tmp = l * (((l * l) * 0.3333333333333333) * J);
	} else {
		tmp = fma((2.0 * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -145000.0) || !(l <= 8e+21))
		tmp = Float64(l * Float64(Float64(Float64(l * l) * 0.3333333333333333) * J));
	else
		tmp = fma(Float64(2.0 * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -145000.0], N[Not[LessEqual[l, 8e+21]], $MachinePrecision]], N[(l * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -145000 or 8e21 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 65.4%

       \[\leadsto \left({\ell}^{3} \cdot 0.3333333333333333\right) \cdot J \]
   12. Step-by-step derivation

   13. Applied rewrites 60.8%

       \[\leadsto \ell \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right) \]

   if -145000 < l < 8e21

   1. Initial program 76.6%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 83.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;\ell \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.5% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right) \cdot \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (fma (* 0.3333333333333333 l) l 2.0) l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((fma((0.3333333333333333 * l), l, 2.0) * l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(fma(Float64(0.3333333333333333 * l), l, 2.0) * l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0

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

5. Applied rewrites 85.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 75.1%

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

10. Applied rewrites 75.1%

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

Alternative 19: 54.4% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0

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

5. Applied rewrites 85.4%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 75.1%

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

9. Taylor expanded in l around 0

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

11. Applied rewrites 57.3%

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

Alternative 20: 36.8% accurate, 330.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in J around 0

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

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025056 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))