Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 99.9%

Time: 6.2s

Alternatives: 17

Speedup: 1.9×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return ((cos((0.5 * K)) * J) * (2.0 * sinh(l))) + 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 = ((cos((0.5d0 * k)) * j) * (2.0d0 * sinh(l))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((Math.cos((0.5 * K)) * J) * (2.0 * Math.sinh(l))) + U;
}

Python

def code(J, l, K, U):
	return ((math.cos((0.5 * K)) * J) * (2.0 * math.sinh(l))) + U

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((cos((0.5 * K)) * J) * (2.0 * sinh(l))) + U;
end

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in J around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sinh-undef N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sinh.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+33}\right):\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+33)))
     (+ (* (* (* (* l l) 0.3333333333333333) l) J) U)
     (+ (* (* l J) 2.0) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 4e+33)) {
		tmp = ((((l * l) * 0.3333333333333333) * l) * J) + U;
	} else {
		tmp = ((l * J) * 2.0) + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 4e+33)) {
		tmp = ((((l * l) * 0.3333333333333333) * l) * J) + U;
	} else {
		tmp = ((l * J) * 2.0) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 4e+33):
		tmp = ((((l * l) * 0.3333333333333333) * l) * J) + U
	else:
		tmp = ((l * J) * 2.0) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 4e+33))
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l) * J) + U);
	else
		tmp = Float64(Float64(Float64(l * J) * 2.0) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 4e+33)))
		tmp = ((((l * l) * 0.3333333333333333) * l) * J) + U;
	else
		tmp = ((l * J) * 2.0) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 4e+33]], $MachinePrecision]], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < -inf.0 or 3.9999999999999998e33 < (+.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 99.9%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 80.6

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

   8. Applied rewrites 80.6%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. fp-cancel-sign-sub-inv N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 69.6

          \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]

   11. Applied rewrites 69.6%

       \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 69.6

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

   14. Applied rewrites 69.6%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 3.9999999999999998e33

   1. Initial program 73.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \ell\right) \cdot 2 + U \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-*.f64 86.4

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

   8. Applied rewrites 86.4%

      \[\leadsto \left(\ell \cdot J\right) \cdot 2 + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq -\infty \lor \neg \left(\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 4 \cdot 10^{+33}\right):\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2 + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* J (- (exp l) (exp (- l)))) -5e-39)
   (* (* (cos (* 0.5 K)) J) (* 2.0 (sinh l)))
   (+
    (*
     (*
      J
      (*
       (+
        (*
         (+
          (*
           (+ (* 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 ((J * (exp(l) - exp(-l))) <= -5e-39) {
		tmp = (cos((0.5 * K)) * J) * (2.0 * sinh(l));
	} else {
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j * (exp(l) - exp(-l))) <= (-5d-39)) then
        tmp = (cos((0.5d0 * k)) * j) * (2.0d0 * sinh(l))
    else
        tmp = ((j * (((((((0.0003968253968253968d0 * (l * l)) + 0.016666666666666666d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l)) * cos((k / 2.0d0))) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J * (Math.exp(l) - Math.exp(-l))) <= -5e-39) {
		tmp = (Math.cos((0.5 * K)) * J) * (2.0 * Math.sinh(l));
	} else {
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * Math.cos((K / 2.0))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J * (math.exp(l) - math.exp(-l))) <= -5e-39:
		tmp = (math.cos((0.5 * K)) * J) * (2.0 * math.sinh(l))
	else:
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * math.cos((K / 2.0))) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(J * Float64(exp(l) - exp(Float64(-l)))) <= -5e-39)
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * J) * Float64(2.0 * sinh(l)));
	else
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(Float64(Float64(Float64(Float64(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

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J * (exp(l) - exp(-l))) <= -5e-39)
		tmp = (cos((0.5 * K)) * J) * (2.0 * sinh(l));
	else
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-39], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $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 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.9999999999999998e-39

   1. Initial program 98.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.9999999999999998e-39 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      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

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

   5. Applied rewrites 98.0%

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

Alternative 4: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.22:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \ell\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.005:\\ \;\;\;\;\left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.22)
     (+ (* (* (cos (* 0.5 K)) J) (* 2.0 l)) U)
     (if (<= t_0 -0.005)
       (+
        (*
         (*
          (- (* (* 0.016666666666666666 (* (* l l) J)) (* l l)) (* -2.0 J))
          l)
         (+ (* -0.125 (* K K)) 1.0))
        U)
       (+ (* (* 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 <= -0.22) {
		tmp = ((cos((0.5 * K)) * J) * (2.0 * l)) + U;
	} else if (t_0 <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((2.0 * sinh(l)) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.22d0)) then
        tmp = ((cos((0.5d0 * k)) * j) * (2.0d0 * l)) + u
    else if (t_0 <= (-0.005d0)) then
        tmp = (((((0.016666666666666666d0 * ((l * l) * j)) * (l * l)) - ((-2.0d0) * j)) * l) * (((-0.125d0) * (k * k)) + 1.0d0)) + u
    else
        tmp = ((2.0d0 * sinh(l)) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.22) {
		tmp = ((Math.cos((0.5 * K)) * J) * (2.0 * l)) + U;
	} else if (t_0 <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((2.0 * Math.sinh(l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.22:
		tmp = ((math.cos((0.5 * K)) * J) * (2.0 * l)) + U
	elif t_0 <= -0.005:
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U
	else:
		tmp = ((2.0 * math.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 <= -0.22)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * Float64(2.0 * l)) + U);
	elseif (t_0 <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.016666666666666666 * Float64(Float64(l * l) * J)) * Float64(l * l)) - Float64(-2.0 * J)) * l) * Float64(Float64(-0.125 * Float64(K * K)) + 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(l)) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.22)
		tmp = ((cos((0.5 * K)) * J) * (2.0 * l)) + U;
	elseif (t_0 <= -0.005)
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	else
		tmp = ((2.0 * sinh(l)) * J) + U;
	end
	tmp_2 = 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, -0.22], N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(N[(N[(0.016666666666666666 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.220000000000000001

   1. Initial program 89.2%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 70.8%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 91.7%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      13. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

      16. lower-*.f64 91.1

          \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   8. Applied rewrites 91.1%

      \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right)} + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left({\ell}^{2} \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       5. lift-*.f64 91.1

          \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   11. Applied rewrites 91.1%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   12. Taylor expanded in K around 0

       \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\frac{-1}{8} \cdot \left(K \cdot K\right) + 1\right) + U \]
   13. Step-by-step derivation

   14. Applied rewrites 91.1%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U \]

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

   1. Initial program 89.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sinh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sinh.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 5: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.8:\\ \;\;\;\;\left(J \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333 + 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.8)
     (+ (* (* J (* (+ (* (* l l) 0.3333333333333333) 2.0) l)) t_0) U)
     (+ (* (* 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 <= 0.8) {
		tmp = ((J * ((((l * l) * 0.3333333333333333) + 2.0) * l)) * t_0) + U;
	} else {
		tmp = ((2.0 * sinh(l)) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.8d0) then
        tmp = ((j * ((((l * l) * 0.3333333333333333d0) + 2.0d0) * l)) * t_0) + u
    else
        tmp = ((2.0d0 * sinh(l)) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.8) {
		tmp = ((J * ((((l * l) * 0.3333333333333333) + 2.0) * l)) * t_0) + U;
	} else {
		tmp = ((2.0 * Math.sinh(l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.8:
		tmp = ((J * ((((l * l) * 0.3333333333333333) + 2.0) * l)) * t_0) + U
	else:
		tmp = ((2.0 * math.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 <= 0.8)
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) + 2.0) * l)) * t_0) + U);
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(l)) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.8)
		tmp = ((J * ((((l * l) * 0.3333333333333333) + 2.0) * l)) * t_0) + U;
	else
		tmp = ((2.0 * sinh(l)) * J) + U;
	end
	tmp_2 = 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, 0.8], N[(N[(N[(J * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.80000000000000004

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if 0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sinh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sinh.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 6: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+
    (*
     (* (- (* (* 0.016666666666666666 (* (* l l) J)) (* l l)) (* -2.0 J)) l)
     (+ (* -0.125 (* K K)) 1.0))
    U)
   (+ (* (* 2.0 (sinh l)) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((2.0 * sinh(l)) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = (((((0.016666666666666666d0 * ((l * l) * j)) * (l * l)) - ((-2.0d0) * j)) * l) * (((-0.125d0) * (k * k)) + 1.0d0)) + u
    else
        tmp = ((2.0d0 * sinh(l)) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((2.0 * Math.sinh(l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U
	else:
		tmp = ((2.0 * math.sinh(l)) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.016666666666666666 * Float64(Float64(l * l) * J)) * Float64(l * l)) - Float64(-2.0 * J)) * l) * Float64(Float64(-0.125 * Float64(K * K)) + 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(l)) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	else
		tmp = ((2.0 * sinh(l)) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(N[(0.016666666666666666 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      13. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

      16. lower-*.f64 51.0

          \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right)} + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left({\ell}^{2} \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       5. lift-*.f64 51.0

          \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   11. Applied rewrites 51.0%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   12. Taylor expanded in K around 0

       \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\frac{-1}{8} \cdot \left(K \cdot K\right) + 1\right) + U \]
   13. Step-by-step derivation

   14. Applied rewrites 60.2%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U \]

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

   1. Initial program 89.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sinh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sinh.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 7: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+
    (*
     (* (- (* (* 0.016666666666666666 (* (* l l) J)) (* l l)) (* -2.0 J)) l)
     (+ (* -0.125 (* K K)) 1.0))
    U)
   (+
    (*
     (*
      (+
       (*
        (+
         (* (+ (* 0.0003968253968253968 (* l l)) 0.016666666666666666) (* l l))
         0.3333333333333333)
        (* l l))
       2.0)
      l)
     J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = (((((0.016666666666666666d0 * ((l * l) * j)) * (l * l)) - ((-2.0d0) * j)) * l) * (((-0.125d0) * (k * k)) + 1.0d0)) + u
    else
        tmp = ((((((((0.0003968253968253968d0 * (l * l)) + 0.016666666666666666d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	} else {
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U
	else:
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.016666666666666666 * Float64(Float64(l * l) * J)) * Float64(l * l)) - Float64(-2.0 * J)) * l) * Float64(Float64(-0.125 * Float64(K * K)) + 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.0003968253968253968 * Float64(l * l)) + 0.016666666666666666) * Float64(l * l)) + 0.3333333333333333) * Float64(l * l)) + 2.0) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = (((((0.016666666666666666 * ((l * l) * J)) * (l * l)) - (-2.0 * J)) * l) * ((-0.125 * (K * K)) + 1.0)) + U;
	else
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(N[(0.016666666666666666 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left({K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + \color{blue}{1}\right) + U \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      13. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) + U \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{3} \cdot J - \frac{-1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

      16. lower-*.f64 51.0

          \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(0.3333333333333333 \cdot J - -0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right)} + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left({\ell}^{2} \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(\frac{-1}{46080} \cdot \left(K \cdot K\right) + \frac{1}{384}\right) \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

       5. lift-*.f64 51.0

          \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   11. Applied rewrites 51.0%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\left(\left(-2.170138888888889 \cdot 10^{-5} \cdot \left(K \cdot K\right) + 0.0026041666666666665\right) \cdot \left(K \cdot K\right) - 0.125\right) \cdot \left(K \cdot K\right) + 1\right) + U \]

   12. Taylor expanded in K around 0

       \[\leadsto \left(\left(\left(\frac{1}{60} \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(\frac{-1}{8} \cdot \left(K \cdot K\right) + 1\right) + U \]
   13. Step-by-step derivation

   14. Applied rewrites 60.2%

       \[\leadsto \left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U \]

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

   1. Initial program 89.7%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 93.5%

       \[\leadsto \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(0.016666666666666666 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \left(\ell \cdot \ell\right) - -2 \cdot J\right) \cdot \ell\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 10^{-79}:\\ \;\;\;\;\left(2 \cdot \sinh \ell\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1e-79)
   (+ (* (* 2.0 (sinh l)) J) U)
   (+
    (*
     (*
      J
      (*
       (+
        (*
         (+
          (*
           (+ (* 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 <= 1e-79) {
		tmp = ((2.0 * sinh(l)) * J) + U;
	} else {
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (k <= 1d-79) then
        tmp = ((2.0d0 * sinh(l)) * j) + u
    else
        tmp = ((j * (((((((0.0003968253968253968d0 * (l * l)) + 0.016666666666666666d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l)) * cos((k / 2.0d0))) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1e-79) {
		tmp = ((2.0 * Math.sinh(l)) * J) + U;
	} else {
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * Math.cos((K / 2.0))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 1e-79:
		tmp = ((2.0 * math.sinh(l)) * J) + U
	else:
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * math.cos((K / 2.0))) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1e-79)
		tmp = Float64(Float64(Float64(2.0 * sinh(l)) * J) + U);
	else
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(Float64(Float64(Float64(Float64(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

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 1e-79)
		tmp = ((2.0 * sinh(l)) * J) + U;
	else
		tmp = ((J * (((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1e-79], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $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 < 1e-79

   1. Initial program 89.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sinh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sinh.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if 1e-79 < K

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0

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

      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

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

   5. Applied rewrites 96.7%

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

Alternative 9: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+
    (*
     (* (- (* 2.0 J) (* -0.3333333333333333 (* (* l l) J))) l)
     (+ (* (* K K) -0.125) 1.0))
    U)
   (+
    (*
     (*
      (+
       (*
        (+
         (* (+ (* 0.0003968253968253968 (* l l)) 0.016666666666666666) (* l l))
         0.3333333333333333)
        (* l l))
       2.0)
      l)
     J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	} else {
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = ((((2.0d0 * j) - ((-0.3333333333333333d0) * ((l * l) * j))) * l) * (((k * k) * (-0.125d0)) + 1.0d0)) + u
    else
        tmp = ((((((((0.0003968253968253968d0 * (l * l)) + 0.016666666666666666d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	} else {
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U
	else:
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * J) - Float64(-0.3333333333333333 * Float64(Float64(l * l) * J))) * l) * Float64(Float64(Float64(K * K) * -0.125) + 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.0003968253968253968 * Float64(l * l)) + 0.016666666666666666) * Float64(l * l)) + 0.3333333333333333) * Float64(l * l)) + 2.0) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	else
		tmp = ((((((((0.0003968253968253968 * (l * l)) + 0.016666666666666666) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(2.0 * J), $MachinePrecision] - N[(-0.3333333333333333 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 57.2

         \[\leadsto \left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U \]

   8. Applied rewrites 57.2%

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

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

   1. Initial program 89.7%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 93.5%

       \[\leadsto \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+
    (*
     (* (- (* 2.0 J) (* -0.3333333333333333 (* (* l l) J))) l)
     (+ (* (* K K) -0.125) 1.0))
    U)
   (+
    (*
     (*
      (+
       (*
        (+ (* (* (* l l) 0.0003968253968253968) (* l l)) 0.3333333333333333)
        (* l l))
       2.0)
      l)
     J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	} else {
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = ((((2.0d0 * j) - ((-0.3333333333333333d0) * ((l * l) * j))) * l) * (((k * k) * (-0.125d0)) + 1.0d0)) + u
    else
        tmp = ((((((((l * l) * 0.0003968253968253968d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	} else {
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U
	else:
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * J) - Float64(-0.3333333333333333 * Float64(Float64(l * l) * J))) * l) * Float64(Float64(Float64(K * K) * -0.125) + 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(l * l) * 0.0003968253968253968) * Float64(l * l)) + 0.3333333333333333) * Float64(l * l)) + 2.0) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = ((((2.0 * J) - (-0.3333333333333333 * ((l * l) * J))) * l) * (((K * K) * -0.125) + 1.0)) + U;
	else
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(2.0 * J), $MachinePrecision] - N[(-0.3333333333333333 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 57.2

         \[\leadsto \left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U \]

   8. Applied rewrites 57.2%

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

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

   1. Initial program 89.7%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 93.5%

       \[\leadsto \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{1}{2520}\right) \cdot \left(\ell \cdot \ell\right) + \frac{1}{3}\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

       4. lift-*.f64 93.2

          \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

   14. Applied rewrites 93.2%

       \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(\left(2 \cdot J - -0.3333333333333333 \cdot \left(\left(\ell \cdot \ell\right) \cdot J\right)\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.746)
   (+ (* (* (* l J) (+ (* (* K K) -0.125) 1.0)) 2.0) U)
   (+
    (*
     (*
      (+
       (*
        (+ (* (* (* l l) 0.0003968253968253968) (* l l)) 0.3333333333333333)
        (* l l))
       2.0)
      l)
     J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.746d0)) then
        tmp = (((l * j) * (((k * k) * (-0.125d0)) + 1.0d0)) * 2.0d0) + u
    else
        tmp = ((((((((l * l) * 0.0003968253968253968d0) * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.746:
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U
	else:
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.746)
		tmp = Float64(Float64(Float64(Float64(l * J) * Float64(Float64(Float64(K * K) * -0.125) + 1.0)) * 2.0) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(l * l) * 0.0003968253968253968) * Float64(l * l)) + 0.3333333333333333) * Float64(l * l)) + 2.0) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.746)
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	else
		tmp = ((((((((l * l) * 0.0003968253968253968) * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.746], N[(N[(N[(N[(l * J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.746

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 55.2

         \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   8. Applied rewrites 55.2%

      \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   if -0.746 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 86.0%

       \[\leadsto \left(\left(\left(\left(0.0003968253968253968 \cdot \left(\ell \cdot \ell\right) + 0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{1}{2520}\right) \cdot \left(\ell \cdot \ell\right) + \frac{1}{3}\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

       4. lift-*.f64 85.8

          \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]

   14. Applied rewrites 85.8%

       \[\leadsto \left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right) \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot -0.125 + 1\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\left(\left(0.016666666666666666 \cdot \left(\ell \cdot \ell\right) + 0.3333333333333333\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2e+20)
   (+ (* (* (+ (* (* K K) -0.125) 1.0) J) (* 2.0 (sinh l))) U)
   (+
    (*
     (*
      J
      (*
       (+
        (* (+ (* 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+20) {
		tmp = (((((K * K) * -0.125) + 1.0) * J) * (2.0 * sinh(l))) + U;
	} else {
		tmp = ((J * (((((0.016666666666666666 * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (k <= 2d+20) then
        tmp = (((((k * k) * (-0.125d0)) + 1.0d0) * j) * (2.0d0 * sinh(l))) + u
    else
        tmp = ((j * (((((0.016666666666666666d0 * (l * l)) + 0.3333333333333333d0) * (l * l)) + 2.0d0) * l)) * cos((k / 2.0d0))) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2e+20) {
		tmp = (((((K * K) * -0.125) + 1.0) * J) * (2.0 * Math.sinh(l))) + U;
	} else {
		tmp = ((J * (((((0.016666666666666666 * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * Math.cos((K / 2.0))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 2e+20:
		tmp = (((((K * K) * -0.125) + 1.0) * J) * (2.0 * math.sinh(l))) + U
	else:
		tmp = ((J * (((((0.016666666666666666 * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * math.cos((K / 2.0))) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2e+20)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(K * K) * -0.125) + 1.0) * J) * Float64(2.0 * sinh(l))) + U);
	else
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(Float64(Float64(0.016666666666666666 * Float64(l * l)) + 0.3333333333333333) * Float64(l * l)) + 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 2e+20)
		tmp = (((((K * K) * -0.125) + 1.0) * J) * (2.0 * sinh(l))) + U;
	else
		tmp = ((J * (((((0.016666666666666666 * (l * l)) + 0.3333333333333333) * (l * l)) + 2.0) * l)) * cos((K / 2.0))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2e+20], N[(N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision]), $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 < 2e20

   1. Initial program 90.7%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 87.6

         \[\leadsto \left(\left(\left(K \cdot K\right) \cdot -0.125 + 1\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right) + U \]

   8. Applied rewrites 87.6%

      \[\leadsto \left(\left(\left(K \cdot K\right) \cdot -0.125 + 1\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right) + U \]

   if 2e20 < K

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      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. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 13: 77.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.3333333333333333 - \left(\ell \cdot \ell\right) \cdot -0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.746)
   (+ (* (* (* l J) (+ (* (* K K) -0.125) 1.0)) 2.0) U)
   (+
    (*
     (*
      (+
       (* (- 0.3333333333333333 (* (* l l) -0.016666666666666666)) (* l l))
       2.0)
      l)
     J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = (((((0.3333333333333333 - ((l * l) * -0.016666666666666666)) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.746d0)) then
        tmp = (((l * j) * (((k * k) * (-0.125d0)) + 1.0d0)) * 2.0d0) + u
    else
        tmp = (((((0.3333333333333333d0 - ((l * l) * (-0.016666666666666666d0))) * (l * l)) + 2.0d0) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = (((((0.3333333333333333 - ((l * l) * -0.016666666666666666)) * (l * l)) + 2.0) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.746:
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U
	else:
		tmp = (((((0.3333333333333333 - ((l * l) * -0.016666666666666666)) * (l * l)) + 2.0) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.746)
		tmp = Float64(Float64(Float64(Float64(l * J) * Float64(Float64(Float64(K * K) * -0.125) + 1.0)) * 2.0) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(l * l) * -0.016666666666666666)) * Float64(l * l)) + 2.0) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.746)
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	else
		tmp = (((((0.3333333333333333 - ((l * l) * -0.016666666666666666)) * (l * l)) + 2.0) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.746], N[(N[(N[(N[(l * J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 - N[(N[(l * l), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.746

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 55.2

         \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   8. Applied rewrites 55.2%

      \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   if -0.746 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

   11. Applied rewrites 84.7%

       \[\leadsto \left(\left(\left(0.3333333333333333 - \left(\ell \cdot \ell\right) \cdot -0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.3333333333333333 - \left(\ell \cdot \ell\right) \cdot -0.016666666666666666\right) \cdot \left(\ell \cdot \ell\right) + 2\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.746)
   (+ (* (* (* l J) (+ (* (* K K) -0.125) 1.0)) 2.0) U)
   (+ (* (* (- 2.0 (* l (* l -0.3333333333333333))) l) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.746d0)) then
        tmp = (((l * j) * (((k * k) * (-0.125d0)) + 1.0d0)) * 2.0d0) + u
    else
        tmp = (((2.0d0 - (l * (l * (-0.3333333333333333d0)))) * l) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.746) {
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	} else {
		tmp = (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.746:
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U
	else:
		tmp = (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.746)
		tmp = Float64(Float64(Float64(Float64(l * J) * Float64(Float64(Float64(K * K) * -0.125) + 1.0)) * 2.0) + U);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 - Float64(l * Float64(l * -0.3333333333333333))) * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.746)
		tmp = (((l * J) * (((K * K) * -0.125) + 1.0)) * 2.0) + U;
	else
		tmp = (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.746], N[(N[(N[(N[(l * J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(2.0 - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.746

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 55.2

         \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   8. Applied rewrites 55.2%

      \[\leadsto \left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U \]

   if -0.746 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in J around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. fp-cancel-sign-sub-inv N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 81.2

          \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]

   11. Applied rewrites 81.2%

       \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 81.2

          \[\leadsto \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \]

   13. Applied rewrites 81.2%

       \[\leadsto \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.746:\\ \;\;\;\;\left(\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125 + 1\right)\right) \cdot 2 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \end{array} \]

FPCore

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

C

double code(double J, double l, double K, double U) {
	return (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + 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 = (((2.0d0 - (l * (l * (-0.3333333333333333d0)))) * l) * j) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U;
}

Python

def code(J, l, K, U):
	return (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = (((2.0 - (l * (l * -0.3333333333333333))) * l) * J) + U;
end

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in J around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sinh-undef N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sinh.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in K around 0

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

   1. sinh-undef-rev N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sinh.f64 83.4

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

8. Applied rewrites 83.4%

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

9. Taylor expanded in l around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. fp-cancel-sign-sub-inv N/A

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

    4. metadata-eval N/A

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

    5. lower--.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 76.3

       \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]

11. Applied rewrites 76.3%

    \[\leadsto \left(\left(2 - \left(\ell \cdot \ell\right) \cdot -0.3333333333333333\right) \cdot \ell\right) \cdot J + U \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 76.3

       \[\leadsto \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \]

13. Applied rewrites 76.3%

    \[\leadsto \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \]

14. Final simplification 76.3%

    \[\leadsto \left(\left(2 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)\right) \cdot \ell\right) \cdot J + U \]
15. Add Preprocessing

Alternative 16: 54.8% accurate, 23.6× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return ((l * J) * 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 = ((l * j) * 2.0d0) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((l * J) * 2.0) + U;
}

Python

def code(J, l, K, U):
	return ((l * J) * 2.0) + U

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in K around 0

   \[\leadsto \left(J \cdot \ell\right) \cdot 2 + U \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lift-*.f64 57.1

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

8. Applied rewrites 57.1%

   \[\leadsto \left(\ell \cdot J\right) \cdot 2 + U \]
9. Add Preprocessing

Alternative 17: 36.8% accurate, 330.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in J around 0

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

5. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))