Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 100.0%

Time: 6.2s

Alternatives: 15

Speedup: 2.1×

• 49.5% 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: 86.5% 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: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 56.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 4e+297)
   (fma (* l J) 2.0 U)
   (fma (* (* l J) (fma (* K K) -0.125 1.0)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 4e+297) {
		tmp = fma((l * J), 2.0, U);
	} else {
		tmp = fma(((l * J) * fma((K * K), -0.125, 1.0)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 4e+297)
		tmp = fma(Float64(l * J), 2.0, U);
	else
		tmp = fma(Float64(Float64(l * J) * fma(Float64(K * K), -0.125, 1.0)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], 4e+297], N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 4.0000000000000001e297

   1. Initial program 81.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

   if 4.0000000000000001e297 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 28.7

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

   4. Applied rewrites 28.7%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

Alternative 3: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.76:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right), 2, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.76)
     (fma (* l (* (cos (* 0.5 K)) J)) 2.0 U)
     (if (<= t_0 -0.01)
       (fma
        J
        (* (* (* (* l l) 0.3333333333333333) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.76) {
		tmp = fma((l * (cos((0.5 * K)) * J)), 2.0, U);
	} else if (t_0 <= -0.01) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * 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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.76)
		tmp = fma(Float64(l * Float64(cos(Float64(0.5 * K)) * J)), 2.0, U);
	elseif (t_0 <= -0.01)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.76], N[(N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.76000000000000001

   1. Initial program 85.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.4

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

   4. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -0.76000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.3

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

   9. Applied rewrites 60.3%

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

   10. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 64.8

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

   12. Applied rewrites 64.8%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

   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 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 4: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.76:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.125, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(0.16666666666666666, \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.76)
     U
     (if (<= t_0 -0.01)
       (fma (* (* (* (* K K) l) J) -0.125) 2.0 U)
       (fma J (* (* (fma 0.16666666666666666 (* l l) 1.0) l) 2.0) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.76) {
		tmp = U;
	} else if (t_0 <= -0.01) {
		tmp = fma(((((K * K) * l) * J) * -0.125), 2.0, U);
	} else {
		tmp = fma(J, ((fma(0.16666666666666666, (l * l), 1.0) * l) * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.76)
		tmp = U;
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.125), 2.0, U);
	else
		tmp = fma(J, Float64(Float64(fma(0.16666666666666666, Float64(l * l), 1.0) * l) * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.76], U, If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.125), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(N[(N[(0.16666666666666666 * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.76000000000000001

   1. Initial program 85.6%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 37.1%

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

   if -0.76000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.8%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 44.9

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

   7. Applied rewrites 44.9%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 56.4

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

   10. Applied rewrites 56.4%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 84.0

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

   9. Applied rewrites 84.0%

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

Alternative 5: 56.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.76:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.125, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.76)
     U
     (if (<= t_0 -0.01)
       (fma (* (* (* (* K K) l) J) -0.125) 2.0 U)
       (fma (* l J) 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.76) {
		tmp = U;
	} else if (t_0 <= -0.01) {
		tmp = fma(((((K * K) * l) * J) * -0.125), 2.0, U);
	} else {
		tmp = fma((l * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.76)
		tmp = U;
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.125), 2.0, U);
	else
		tmp = fma(Float64(l * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.76], U, If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.125), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(l * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.76000000000000001

   1. Initial program 85.6%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 37.1%

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

   if -0.76000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.8%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 44.9

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

   7. Applied rewrites 44.9%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 56.4

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

   10. Applied rewrites 56.4%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

   2. Taylor expanded in l around 0

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

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.7

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 59.7

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

   7. Applied rewrites 59.7%

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

Alternative 6: 88.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma J (* (* (* (* l l) 0.3333333333333333) 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)) <= -0.01) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * 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)) <= -0.01)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * 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], -0.01], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(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))) < -0.0100000000000000002

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

   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 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 7: 100.0% accurate, 1.5× 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 86.5%

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

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 100.0

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

4. Applied rewrites 100.0%

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

Alternative 8: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2e-113)
   (fma (* 2.0 (sinh l)) J U)
   (+
    (*
     (*
      J
      (*
       (fma
        (fma
         (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2e-113) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2e-113)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2e-113], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.99999999999999996e-113

   1. Initial program 86.1%

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

   2. Taylor expanded in 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 85.6

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

   4. Applied rewrites 85.6%

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

   if 1.99999999999999996e-113 < K

   1. Initial program 87.2%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.8%

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

Alternative 9: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma J (* (* (* (* l l) 0.3333333333333333) l) (fma (* K K) -0.125 1.0)) U)
   (fma
    J
    (*
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* l l) 0.008333333333333333)
        (* l l)
        0.16666666666666666)
       (* l l)
       1.0)
      l)
     2.0)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, ((fma(fma(fma(0.0001984126984126984, (l * l), 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l) * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(l * l), 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(N[(0.0001984126984126984 * N[(l * l), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot 2, U\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right) \cdot 2, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right) \cdot 2, U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2} + 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left({\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      16. lift-*.f64 91.0

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

   9. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2e-89)
   (fma (* 2.0 (sinh l)) J U)
   (+
    (*
     (*
      J
      (*
       (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2e-89) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2e-89)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2e-89], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.00000000000000008e-89

   1. Initial program 86.1%

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

   2. Taylor expanded in 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 86.0

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

   4. Applied rewrites 86.0%

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

   if 2.00000000000000008e-89 < K

   1. Initial program 87.2%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 11: 83.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma J (* (* (* (* l l) 0.3333333333333333) l) (fma (* K K) -0.125 1.0)) U)
   (fma
    J
    (*
     (*
      (fma (fma 0.008333333333333333 (* l l) 0.16666666666666666) (* l l) 1.0)
      l)
     2.0)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, ((fma(fma(0.008333333333333333, (l * l), 0.16666666666666666), (l * l), 1.0) * l) * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(Float64(fma(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)\right) \cdot 2, U\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right) \cdot 2, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right) \cdot 2, U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) + 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{2} + 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

      11. lift-*.f64 89.0

          \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]

   9. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(J, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(0.16666666666666666, \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma J (* (* (* (* l l) 0.3333333333333333) l) (fma (* K K) -0.125 1.0)) U)
   (fma J (* (* (fma 0.16666666666666666 (* l l) 1.0) l) 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(J, ((((l * l) * 0.3333333333333333) * l) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(J, ((fma(0.16666666666666666, (l * l), 1.0) * l) * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(J, Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(J, Float64(Float64(fma(0.16666666666666666, Float64(l * l), 1.0) * l) * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(0.16666666666666666 * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 84.0

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

   9. Applied rewrites 84.0%

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

Alternative 13: 86.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2.8e-88)
   (fma (* 2.0 (sinh l)) J U)
   (fma J (* (* (fma (* l l) 0.3333333333333333 2.0) l) (cos (* 0.5 K))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.8e-88) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = fma(J, ((fma((l * l), 0.3333333333333333, 2.0) * l) * cos((0.5 * K))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2.8e-88)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = fma(J, Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * cos(Float64(0.5 * K))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2.8e-88], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.79999999999999976e-88

   1. Initial program 86.1%

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

   2. Taylor expanded in 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 86.0

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

   4. Applied rewrites 86.0%

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

   if 2.79999999999999976e-88 < K

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 87.3

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

   9. Applied rewrites 87.3%

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

Alternative 14: 54.1% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 63.7

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

4. Applied rewrites 63.7%

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

5. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lift-*.f64 54.1

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

7. Applied rewrites 54.1%

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

Alternative 15: 37.2% accurate, 330.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2025092 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))