Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.9%

Time: 4.9s

Alternatives: 12

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sinh.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-sinh.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \left(\left(J + J\right) \cdot t\_0\right) \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -1.9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J \cdot \left(\ell + \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (* (* (+ J J) t_0) (sinh l))))
   (if (<= l -1.9) t_1 (if (<= l 0.22) (fma t_0 (* J (+ l l)) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = ((J + J) * t_0) * sinh(l);
	double tmp;
	if (l <= -1.9) {
		tmp = t_1;
	} else if (l <= 0.22) {
		tmp = fma(t_0, (J * (l + l)), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(Float64(Float64(J + J) * t_0) * sinh(l))
	tmp = 0.0
	if (l <= -1.9)
		tmp = t_1;
	elseif (l <= 0.22)
		tmp = fma(t_0, Float64(J * Float64(l + l)), U);
	else
		tmp = t_1;
	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] * t$95$0), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9], t$95$1, If[LessEqual[l, 0.22], N[(t$95$0 * N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8999999999999999 or 0.220000000000000001 < l

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. sinh-undef-rev N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sinh.f64 65.8

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

   7. Applied rewrites 65.8%

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

   if -1.8999999999999999 < l < 0.220000000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 64.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 64.4

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 64.4

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

   9. Applied rewrites 64.4%

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

Alternative 3: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.35:\\ \;\;\;\;U \cdot \mathsf{fma}\left(J, \frac{\left(\ell + \ell\right) \cdot \cos \left(0.5 \cdot K\right)}{U}, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.0325:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot -0.25\right) \cdot J\right) \cdot \sinh \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.35)
     (* U (fma J (/ (* (+ l l) (cos (* 0.5 K))) U) 1.0))
     (if (<= t_0 -0.0325)
       (* (* (* (* K K) -0.25) J) (sinh l))
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.35) {
		tmp = U * fma(J, (((l + l) * cos((0.5 * K))) / U), 1.0);
	} else if (t_0 <= -0.0325) {
		tmp = (((K * K) * -0.25) * J) * sinh(l);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.35)
		tmp = Float64(U * fma(J, Float64(Float64(Float64(l + l) * cos(Float64(0.5 * K))) / U), 1.0));
	elseif (t_0 <= -0.0325)
		tmp = Float64(Float64(Float64(Float64(K * K) * -0.25) * J) * sinh(l));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.35], N[(U * N[(J * N[(N[(N[(l + l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.0325], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision] * J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.34999999999999998

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 97.9%

      \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(J, \frac{\cos \left(0.5 \cdot K\right) \cdot \left(\sinh \ell \cdot 2\right)}{U}, 1\right)} \]

   8. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 72.2

         \[\leadsto U \cdot \mathsf{fma}\left(J, \frac{\left(\ell + \ell\right) \cdot \cos \left(0.5 \cdot K\right)}{U}, 1\right) \]

   10. Applied rewrites 72.2%

       \[\leadsto U \cdot \mathsf{fma}\left(J, \frac{\left(\ell + \ell\right) \cdot \cos \left(0.5 \cdot K\right)}{U}, 1\right) \]

   if -0.34999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. sinh-undef-rev N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sinh.f64 65.8

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 46.3

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

   10. Applied rewrites 46.3%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 14.3

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

   13. Applied rewrites 14.3%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 4: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.463:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.463)
     (fma t_0 (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) U)
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.463) {
		tmp = fma(t_0, ((fma((l * l), 0.3333333333333333, 2.0) * l) * J), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.463)
		tmp = fma(t_0, Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.463], N[(t$95$0 * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.463000000000000023

   1. Initial program 86.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 88.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.2

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

      8. sinh-undef-rev 88.2

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

   6. Applied rewrites 88.2%

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

   if 0.463000000000000023 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 5: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.0325:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.0325)
   (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.0325) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.0325)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.0325], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 6: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.0325:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.125, J\right), t\_0, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= (cos (/ K 2.0)) -0.0325)
     (fma (fma (* (* K K) J) -0.125 J) t_0 U)
     (fma t_0 J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (cos((K / 2.0)) <= -0.0325) {
		tmp = fma(fma(((K * K) * J), -0.125, J), t_0, U);
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.0325)
		tmp = fma(fma(Float64(Float64(K * K) * J), -0.125, J), t_0, U);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.0325], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.125 + J), $MachinePrecision] * t$95$0 + U), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 68.7

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

   7. Applied rewrites 68.7%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 7: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.0325:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.125, J\right), \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.0325)
   (fma
    (fma (* (* K K) J) -0.125 J)
    (* (fma (* l l) 0.3333333333333333 2.0) l)
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.0325) {
		tmp = fma(fma(((K * K) * J), -0.125, J), (fma((l * l), 0.3333333333333333, 2.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.0325)
		tmp = fma(fma(Float64(Float64(K * K) * J), -0.125, J), Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.0325], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.125 + J), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 68.7

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.7

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

   10. Applied rewrites 61.7%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 8: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.0325:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.125, \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.0325)
   (fma (* (* (* K K) J) -0.125) (+ l l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.0325) {
		tmp = fma((((K * K) * J) * -0.125), (l + l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.0325)
		tmp = fma(Float64(Float64(Float64(K * K) * J) * -0.125), Float64(l + l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.0325], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.125), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 68.7

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 33.9

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

   13. Applied rewrites 33.9%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 9: 80.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{if}\;\ell \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 215:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, 1 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (sinh l) 2.0) J)))
   (if (<= l -2.7) t_0 (if (<= l 215.0) (fma (+ l l) (* 1.0 J) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (sinh(l) * 2.0) * J;
	double tmp;
	if (l <= -2.7) {
		tmp = t_0;
	} else if (l <= 215.0) {
		tmp = fma((l + l), (1.0 * J), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(sinh(l) * 2.0) * J)
	tmp = 0.0
	if (l <= -2.7)
		tmp = t_0;
	elseif (l <= 215.0)
		tmp = fma(Float64(l + l), Float64(1.0 * J), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -2.7], t$95$0, If[LessEqual[l, 215.0], N[(N[(l + l), $MachinePrecision] * N[(1.0 * J), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.7000000000000002 or 215 < l

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. sinh-undef-rev N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sinh.f64 65.8

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. sinh-undef-rev N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 46.7

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

   10. Applied rewrites 46.7%

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

   if -2.7000000000000002 < l < 215

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 54.3%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 54.3

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

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

       6. lift-+.f64 54.3

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

   12. Applied rewrites 54.3%

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

Alternative 10: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.0325:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot J\right) \cdot -0.125, \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, 1 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.0325)
   (fma (* (* (* K K) J) -0.125) (+ l l) U)
   (fma (+ l l) (* 1.0 J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.0325) {
		tmp = fma((((K * K) * J) * -0.125), (l + l), U);
	} else {
		tmp = fma((l + l), (1.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.0325)
		tmp = fma(Float64(Float64(Float64(K * K) * J) * -0.125), Float64(l + l), U);
	else
		tmp = fma(Float64(l + l), Float64(1.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.0325], N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.125), $MachinePrecision] * N[(l + l), $MachinePrecision] + U), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * N[(1.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.032500000000000001

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 68.7

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 33.9

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

   13. Applied rewrites 33.9%

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

   if -0.032500000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 54.3%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 54.3

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

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

       6. lift-+.f64 54.3

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

   12. Applied rewrites 54.3%

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

Alternative 11: 54.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell + \ell, 1 \cdot J, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (+ l l) (* 1.0 J) U))

C

double code(double J, double l, double K, double U) {
	return fma((l + l), (1.0 * J), U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(l + l), Float64(1.0 * J), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(l + l), $MachinePrecision] * N[(1.0 * J), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in l around 0

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

7. Applied rewrites 64.4%

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

8. Taylor expanded in K around 0

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

10. Applied rewrites 54.3%

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

    1. lift-fma.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 54.3

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

    4. lift-*.f64 N/A

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

    5. count-2-rev N/A

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

    6. lift-+.f64 54.3

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

12. Applied rewrites 54.3%

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

Alternative 12: 36.3% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.3%

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

Reproduce

herbie shell --seed 2025138 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))