Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.9%

Time: 5.3s

Alternatives: 18

Speedup: 1.2×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. sinh-undef N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sinh.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-/.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos \left(0.5 \cdot K\right) \cdot \left(\sinh \ell \cdot 2\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -0.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(\frac{K}{2}\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (cos (* 0.5 K)) (* (sinh l) 2.0)) J)))
   (if (<= l -0.45)
     t_0
     (if (<= l 2.6e-5)
       (fma (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) (cos (/ K 2.0)) U)
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (cos((0.5 * K)) * (sinh(l) * 2.0)) * J;
	double tmp;
	if (l <= -0.45) {
		tmp = t_0;
	} else if (l <= 2.6e-5) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), cos((K / 2.0)), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(cos(Float64(0.5 * K)) * Float64(sinh(l) * 2.0)) * J)
	tmp = 0.0
	if (l <= -0.45)
		tmp = t_0;
	elseif (l <= 2.6e-5)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), cos(Float64(K / 2.0)), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -0.45], t$95$0, If[LessEqual[l, 2.6e-5], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.450000000000000011 or 2.59999999999999984e-5 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in J around inf

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. rec-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sinh-undef-rev N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sinh.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if -0.450000000000000011 < l < 2.59999999999999984e-5

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $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. Add Preprocessing

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \left(t\_0 \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -130:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(J + J, t\_0 \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(\sinh \ell \cdot J\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (* (* t_0 (* (fma (* l l) 0.3333333333333333 2.0) l)) J)))
   (if (<= l -2.4e+123)
     t_1
     (if (<= l -130.0)
       (* (* (sinh l) 2.0) J)
       (if (<= l 2.6e-5)
         (fma (+ J J) (* t_0 l) U)
         (if (<= l 8.5e+74)
           (fma (* 2.0 (* (sinh l) J)) (fma (* K K) -0.125 1.0) U)
           t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = (t_0 * (fma((l * l), 0.3333333333333333, 2.0) * l)) * J;
	double tmp;
	if (l <= -2.4e+123) {
		tmp = t_1;
	} else if (l <= -130.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 2.6e-5) {
		tmp = fma((J + J), (t_0 * l), U);
	} else if (l <= 8.5e+74) {
		tmp = fma((2.0 * (sinh(l) * J)), fma((K * K), -0.125, 1.0), 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(t_0 * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * J)
	tmp = 0.0
	if (l <= -2.4e+123)
		tmp = t_1;
	elseif (l <= -130.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 2.6e-5)
		tmp = fma(Float64(J + J), Float64(t_0 * l), U);
	elseif (l <= 8.5e+74)
		tmp = fma(Float64(2.0 * Float64(sinh(l) * J)), fma(Float64(K * K), -0.125, 1.0), 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[(t$95$0 * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -2.4e+123], t$95$1, If[LessEqual[l, -130.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 2.6e-5], N[(N[(J + J), $MachinePrecision] * N[(t$95$0 * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 8.5e+74], N[(N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.39999999999999989e123 or 8.50000000000000028e74 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in J around inf

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. rec-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sinh-undef-rev N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sinh.f64 65.1

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

   6. Applied rewrites 65.1%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   if -2.39999999999999989e123 < l < -130

   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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 55.5

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

   7. Applied rewrites 55.5%

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

   8. Taylor expanded in J around inf

      \[\leadsto \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot J \]
   9. Step-by-step derivation

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lift-sinh.f64 N/A

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

      4. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if -130 < l < 2.59999999999999984e-5

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.2

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

   4. Applied rewrites 64.2%

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

   if 2.59999999999999984e-5 < l < 8.50000000000000028e74

   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 63.9

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. sinh-undef-rev N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 68.7

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

   6. Applied rewrites 68.7%

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

Alternative 5: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.978:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, t\_0, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.978)
     (fma (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0 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 <= 0.978) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), t_0, 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 <= 0.978)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), t_0, U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.978], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0 + 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))) < 0.97799999999999998

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

   if 0.97799999999999998 < (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 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 6: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \mathsf{fma}\left(2 \cdot \left(\sinh \ell \cdot J\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l)))))
        (t_1 (fma (* 2.0 (* (sinh l) J)) (fma (* K K) -0.125 1.0) U)))
   (if (<= t_0 -4e+227)
     t_1
     (if (<= t_0 0.0) (fma (+ J J) (* (cos (* 0.5 K)) l) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = fma((2.0 * (sinh(l) * J)), fma((K * K), -0.125, 1.0), U);
	double tmp;
	if (t_0 <= -4e+227) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = fma(Float64(2.0 * Float64(sinh(l) * J)), fma(Float64(K * K), -0.125, 1.0), U)
	tmp = 0.0
	if (t_0 <= -4e+227)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+227], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.0000000000000004e227 or 0.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 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 63.9

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. sinh-undef-rev N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 68.7

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

   6. Applied rewrites 68.7%

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

   if -4.0000000000000004e227 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 0.0

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.2

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

   4. Applied rewrites 64.2%

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

Alternative 7: 87.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ (* (* 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)) <= -0.005) {
		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)) <= -0.005)
		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], -0.005], 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))) < -0.0050000000000000001

   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 63.9

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

   4. Applied rewrites 63.9%

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

   if -0.0050000000000000001 < (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 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 8: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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 63.9

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. sinh-undef-rev N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 68.7

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

   6. Applied rewrites 68.7%

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

   7. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 37.4

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

   9. Applied rewrites 37.4%

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

   if -0.0050000000000000001 < (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 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 9: 85.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (+ J J) (* (fma (* K K) -0.125 1.0) 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)) <= -0.005) {
		tmp = fma((J + J), (fma((K * K), -0.125, 1.0) * 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)) <= -0.005)
		tmp = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * 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], -0.005], N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $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))) < -0.0050000000000000001

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if -0.0050000000000000001 < (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 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 10: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(\mathsf{fma}\left(0.16666666666666666, \ell \cdot \ell, 1\right) \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)
   (fma (* 2.0 (* (fma 0.16666666666666666 (* l l) 1.0) l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	} else {
		tmp = fma((2.0 * (fma(0.16666666666666666, (l * l), 1.0) * l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U);
	else
		tmp = fma(Float64(2.0 * Float64(fma(0.16666666666666666, Float64(l * l), 1.0) * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(N[(0.16666666666666666 * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if -0.0050000000000000001 < (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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.4

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

   7. Applied rewrites 71.4%

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

Alternative 11: 79.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if -0.0050000000000000001 < (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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 69.3

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

   7. Applied rewrites 69.3%

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

Alternative 12: 76.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{if}\;\ell \leq -130:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.3333333333333333, J + J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (sinh l) 2.0) J)))
   (if (<= l -130.0)
     t_0
     (if (<= l 2.6e-5)
       (fma (fma (* (* l l) J) 0.3333333333333333 (+ J J)) l 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 <= -130.0) {
		tmp = t_0;
	} else if (l <= 2.6e-5) {
		tmp = fma(fma(((l * l) * J), 0.3333333333333333, (J + J)), l, 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 <= -130.0)
		tmp = t_0;
	elseif (l <= 2.6e-5)
		tmp = fma(fma(Float64(Float64(l * l) * J), 0.3333333333333333, Float64(J + J)), l, 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, -130.0], t$95$0, If[LessEqual[l, 2.6e-5], N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333 + N[(J + J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -130 or 2.59999999999999984e-5 < 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 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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 55.5

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

   7. Applied rewrites 55.5%

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

   8. Taylor expanded in J around inf

      \[\leadsto \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot J \]
   9. Step-by-step derivation

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lift-sinh.f64 N/A

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

      4. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if -130 < l < 2.59999999999999984e-5

   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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 69.3

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

   7. Applied rewrites 69.3%

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

Alternative 13: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* (* (sinh l) 2.0) J)))
   (if (<= t_0 -4e+227) t_1 (if (<= t_0 5e-90) (fma (+ J J) l U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = (sinh(l) * 2.0) * J;
	double tmp;
	if (t_0 <= -4e+227) {
		tmp = t_1;
	} else if (t_0 <= 5e-90) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(Float64(sinh(l) * 2.0) * J)
	tmp = 0.0
	if (t_0 <= -4e+227)
		tmp = t_1;
	elseif (t_0 <= 5e-90)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+227], t$95$1, If[LessEqual[t$95$0, 5e-90], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.0000000000000004e227 or 5.00000000000000019e-90 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 55.5

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

   7. Applied rewrites 55.5%

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

   8. Taylor expanded in J around inf

      \[\leadsto \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot J \]
   9. Step-by-step derivation

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lift-sinh.f64 N/A

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

      4. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if -4.0000000000000004e227 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 5.00000000000000019e-90

   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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   7. Applied rewrites 54.1%

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

Alternative 14: 70.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot 0.16666666666666666\right), J, U\right)\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1350000000000:\\ \;\;\;\;\mathsf{fma}\left(J, \frac{\ell + \ell}{U}, 1\right) \cdot U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* 2.0 (* (* (* l l) l) 0.16666666666666666)) J U)))
   (if (<= l -3.9e+93)
     t_0
     (if (<= l 1350000000000.0) (* (fma J (/ (+ l l) U) 1.0) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((2.0 * (((l * l) * l) * 0.16666666666666666)), J, U);
	double tmp;
	if (l <= -3.9e+93) {
		tmp = t_0;
	} else if (l <= 1350000000000.0) {
		tmp = fma(J, ((l + l) / U), 1.0) * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(2.0 * Float64(Float64(Float64(l * l) * l) * 0.16666666666666666)), J, U)
	tmp = 0.0
	if (l <= -3.9e+93)
		tmp = t_0;
	elseif (l <= 1350000000000.0)
		tmp = Float64(fma(J, Float64(Float64(l + l) / U), 1.0) * U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]}, If[LessEqual[l, -3.9e+93], t$95$0, If[LessEqual[l, 1350000000000.0], N[(N[(J * N[(N[(l + l), $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.9000000000000002e93 or 1.35e12 < 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 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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.4

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

   7. Applied rewrites 71.4%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left({\ell}^{3} \cdot \frac{1}{6}\right), J, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left({\ell}^{3} \cdot \frac{1}{6}\right), J, U\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -3.9000000000000002e93 < l < 1.35e12

   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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. rec-exp N/A

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

      7. sinh-undef-rev N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lower-/.f64 79.1

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

      11. lift-*.f64 N/A

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

      12. lift-sinh.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-sinh.f64 N/A

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

      15. lift-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in l around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 60.5

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

   10. Applied rewrites 60.5%

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

Alternative 15: 60.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := N[(N[(J * N[(N[(l + l), $MachinePrecision] / U), $MachinePrecision] + 1.0), $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 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 80.1

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in U around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. rec-exp N/A

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

   7. sinh-undef-rev N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lower-/.f64 79.1

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

   11. lift-*.f64 N/A

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

   12. lift-sinh.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-sinh.f64 N/A

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

   15. lift-*.f64 79.1

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

7. Applied rewrites 79.1%

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

8. Taylor expanded in l around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f64 60.5

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

10. Applied rewrites 60.5%

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

Alternative 16: 54.1% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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 80.1

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-+.f64 54.1

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

7. Applied rewrites 54.1%

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

Alternative 17: 46.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \left(J + J\right) \cdot \ell\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-90}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* (+ J J) l)))
   (if (<= t_0 -4e+227) t_1 (if (<= t_0 5e-90) U t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = (J + J) * l;
	double tmp;
	if (t_0 <= -4e+227) {
		tmp = t_1;
	} else if (t_0 <= 5e-90) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    t_1 = (j + j) * l
    if (t_0 <= (-4d+227)) then
        tmp = t_1
    else if (t_0 <= 5d-90) then
        tmp = u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double t_1 = (J + J) * l;
	double tmp;
	if (t_0 <= -4e+227) {
		tmp = t_1;
	} else if (t_0 <= 5e-90) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	t_1 = (J + J) * l
	tmp = 0
	if t_0 <= -4e+227:
		tmp = t_1
	elif t_0 <= 5e-90:
		tmp = U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(Float64(J + J) * l)
	tmp = 0.0
	if (t_0 <= -4e+227)
		tmp = t_1;
	elseif (t_0 <= 5e-90)
		tmp = U;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	t_1 = (J + J) * l;
	tmp = 0.0;
	if (t_0 <= -4e+227)
		tmp = t_1;
	elseif (t_0 <= 5e-90)
		tmp = U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+227], t$95$1, If[LessEqual[t$95$0, 5e-90], U, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.0000000000000004e227 or 5.00000000000000019e-90 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 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 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 19.7

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

   10. Applied rewrites 19.7%

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

   if -4.0000000000000004e227 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 5.00000000000000019e-90

   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 36.9%

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

Alternative 18: 36.9% 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 36.9%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))