Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 5.5s

Alternatives: 18

Speedup: 1.5×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 46.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell + \ell\right)\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (+ l l)))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 (- INFINITY)) t_0 (if (<= t_1 2e+42) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l + l);
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 2e+42) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l + l);
	double t_1 = (J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= 2e+42) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l + l)
	t_1 = (J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= 2e+42:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l + l))
	t_1 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 2e+42)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l + l);
	t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= 2e+42)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 2e+42], U, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -inf.0 or 2.00000000000000009e42 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.8%

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

   2. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   3. 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 22.7%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 22.7

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

   12. Applied rewrites 22.7%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.00000000000000009e42

   1. Initial program 72.4%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 71.0%

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

Alternative 3: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := J \cdot \left(2 \cdot \left(\mathsf{fma}\left(0.16666666666666666, \ell \cdot \ell, 1\right) \cdot \ell\right)\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(J \cdot \left(2 \cdot \ell\right)\right) \cdot 1 + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l)))))
        (t_1 (* J (* 2.0 (* (fma 0.16666666666666666 (* l l) 1.0) l)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 2e+280) (+ (* (* J (* 2.0 l)) 1.0) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = J * (2.0 * (fma(0.16666666666666666, (l * l), 1.0) * l));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 2e+280) {
		tmp = ((J * (2.0 * l)) * 1.0) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(J * Float64(2.0 * Float64(fma(0.16666666666666666, Float64(l * l), 1.0) * l)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 2e+280)
		tmp = Float64(Float64(Float64(J * Float64(2.0 * l)) * 1.0) + U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(2.0 * N[(N[(0.16666666666666666 * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 2e+280], N[(N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0 or 2.0000000000000001e280 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.9%

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

   2. Taylor expanded in J around inf

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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 58.3

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

   10. Applied rewrites 58.3%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.0000000000000001e280

   1. Initial program 72.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot 1 + U \]
   9. Step-by-step derivation

   10. Applied rewrites 85.6%

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

Alternative 4: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.97], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 5: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.97], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 6: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) (cos (* 0.5 K)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.97) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), cos((0.5 * K)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.97)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), cos(Float64(0.5 * K)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.97], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 87.6

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

   7. Applied rewrites 87.6%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)} + U \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 87.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.6

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

      7. sinh-undef-rev 87.6

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

   9. Applied rewrites 87.6%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 7: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma J (* (* l 2.0) (cos (* 0.5 K))) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(J, ((l * 2.0) * cos((0.5 * K))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(J, Float64(Float64(l * 2.0) * cos(Float64(0.5 * K))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(J * N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 8: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (* (* l J) (cos (* 0.5 K))) 2.0 U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma(((l * J) * cos((0.5 * K))), 2.0, U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(Float64(l * J) * cos(Float64(0.5 * K))), 2.0, U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(l * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.8%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 9: 87.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.8%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.0

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   7. Applied rewrites 61.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)} + U \]

   8. Taylor expanded in J around 0

      \[\leadsto \left(\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.2

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   10. Applied rewrites 61.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

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

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 10: 96.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 4e-16)
   (+
    (*
     (*
      J
      (*
       (fma
        (fma
         (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       l))
     (cos (/ K 2.0)))
    U)
   (* (* (cos (* 0.5 K)) J) (* 2.0 (sinh l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e-16) {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	} else {
		tmp = (cos((0.5 * K)) * J) * (2.0 * sinh(l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 4e-16)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * J) * Float64(2.0 * sinh(l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 4e-16], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.9999999999999999e-16

   1. Initial program 81.6%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

   if 3.9999999999999999e-16 < l

   1. Initial program 99.2%

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

   2. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   3. 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 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 11: 82.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] + N[(0.016666666666666666 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(0.3333333333333333 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 1.0), $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 85.8%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.0

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   7. Applied rewrites 61.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)} + U \]

   8. Taylor expanded in J around 0

      \[\leadsto \left(\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.2

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   10. Applied rewrites 61.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

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

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 89.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666 \cdot J\right), \ell \cdot \ell, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right)} \cdot 1 + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.016666666666666666 + N[(0.3333333333333333 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 1.0), $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 85.8%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.0

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   7. Applied rewrites 61.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)} + U \]

   8. Taylor expanded in J around 0

      \[\leadsto \left(\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.2

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   10. Applied rewrites 61.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

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

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 86.9%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{1} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.016666666666666666 + N[(0.3333333333333333 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 1.0), $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 85.8%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 86.9%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{1} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 78.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 1.0), $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 85.8%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 84.7%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \color{blue}{1} + U \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 84.7

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

   9. Applied rewrites 84.7%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot 1 + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ (* (* (* 2.0 J) l) (fma (* K K) -0.125 1.0)) U)
   (+ (* (* J (* (fma l (* l 0.3333333333333333) 2.0) l)) 1.0) 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) * l) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = ((J * (fma(l, (l * 0.3333333333333333), 2.0) * l)) * 1.0) + 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(2.0 * J) * l) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * l)) * 1.0) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(N[(2.0 * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 1.0), $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 85.8%

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

   2. Taylor expanded in l around 0

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.0

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   7. Applied rewrites 61.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)} + U \]

   8. Taylor expanded in l around 0

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

      1. lift-*.f64 52.4

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

   10. Applied rewrites 52.4%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 84.7%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \color{blue}{1} + U \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 84.7

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

   9. Applied rewrites 84.7%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot 1 + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 77.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+83}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;\ell \leq 350:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666 \cdot J\right), \ell \cdot \ell, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot 1 + U\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+77}:\\ \;\;\;\;J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665 \cdot J\right), K \cdot K, -0.125 \cdot J\right), K \cdot K, J\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.2e+83)
   (+
    (*
     (* J (* (fma (* l l) 0.3333333333333333 2.0) l))
     (fma (* K K) -0.125 1.0))
    U)
   (if (<= l 350.0)
     (+
      (*
       (*
        (fma
         (fma
          (fma 0.0003968253968253968 (* (* l l) J) (* 0.016666666666666666 J))
          (* l l)
          (* 0.3333333333333333 J))
         (* l l)
         (* 2.0 J))
        l)
       1.0)
      U)
     (if (<= l 2e+77)
       (* J (* 2.0 (sinh l)))
       (*
        (fma
         (fma
          (fma (* (* K K) J) -2.170138888888889e-5 (* 0.0026041666666666665 J))
          (* K K)
          (* -0.125 J))
         (* K K)
         J)
        (*
         2.0
         (*
          (fma
           (fma 0.008333333333333333 (* l l) 0.16666666666666666)
           (* l l)
           1.0)
          l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e+83) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * fma((K * K), -0.125, 1.0)) + U;
	} else if (l <= 350.0) {
		tmp = ((fma(fma(fma(0.0003968253968253968, ((l * l) * J), (0.016666666666666666 * J)), (l * l), (0.3333333333333333 * J)), (l * l), (2.0 * J)) * l) * 1.0) + U;
	} else if (l <= 2e+77) {
		tmp = J * (2.0 * sinh(l));
	} else {
		tmp = fma(fma(fma(((K * K) * J), -2.170138888888889e-5, (0.0026041666666666665 * J)), (K * K), (-0.125 * J)), (K * K), J) * (2.0 * (fma(fma(0.008333333333333333, (l * l), 0.16666666666666666), (l * l), 1.0) * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.2e+83)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (l <= 350.0)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(Float64(l * l) * J), Float64(0.016666666666666666 * J)), Float64(l * l), Float64(0.3333333333333333 * J)), Float64(l * l), Float64(2.0 * J)) * l) * 1.0) + U);
	elseif (l <= 2e+77)
		tmp = Float64(J * Float64(2.0 * sinh(l)));
	else
		tmp = Float64(fma(fma(fma(Float64(Float64(K * K) * J), -2.170138888888889e-5, Float64(0.0026041666666666665 * J)), Float64(K * K), Float64(-0.125 * J)), Float64(K * K), J) * Float64(2.0 * Float64(fma(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.2e+83], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 350.0], N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] + N[(0.016666666666666666 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(0.3333333333333333 * J), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 2e+77], N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -2.170138888888889e-5 + N[(0.0026041666666666665 * J), $MachinePrecision]), $MachinePrecision] * N[(K * K), $MachinePrecision] + N[(-0.125 * J), $MachinePrecision]), $MachinePrecision] * N[(K * K), $MachinePrecision] + J), $MachinePrecision] * N[(2.0 * N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.20000000000000005e83

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.0

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

   7. Applied rewrites 75.0%

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

   if -4.20000000000000005e83 < l < 350

   1. Initial program 75.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 80.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \left(\ell \cdot \ell\right) \cdot J, 0.016666666666666666 \cdot J\right), \ell \cdot \ell, 0.3333333333333333 \cdot J\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right)} \cdot 1 + U \]

   if 350 < l < 1.99999999999999997e77

   1. Initial program 99.8%

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

   2. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 70.2%

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

   if 1.99999999999999997e77 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in l around 0

      \[\leadsto J \cdot \left(2 \cdot \left(\ell \cdot \color{blue}{\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 76.0

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

   10. Applied rewrites 76.0%

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

   11. Taylor expanded in K around 0

       \[\leadsto \left(J + {K}^{2} \cdot \left(\frac{-1}{8} \cdot J + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{384} \cdot J\right)\right)\right) \cdot \left(\color{blue}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({K}^{2} \cdot \left(\frac{-1}{8} \cdot J + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{384} \cdot J\right)\right) + J\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot J + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{384} \cdot J\right)\right) \cdot {K}^{2} + J\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot J + {K}^{2} \cdot \left(\frac{-1}{46080} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{384} \cdot J\right), {K}^{2}, J\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \]

   13. Applied rewrites 73.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -2.170138888888889 \cdot 10^{-5}, 0.0026041666666666665 \cdot J\right), K \cdot K, -0.125 \cdot J\right), K \cdot K, J\right) \cdot \left(\color{blue}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 54.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot 1 + U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ (* (* J (* 2.0 l)) 1.0) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (2.0 * l)) * 1.0) + 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 * (2.0d0 * l)) * 1.0d0) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (2.0 * l)) * 1.0) + U;
}

Python

def code(J, l, K, U):
	return ((J * (2.0 * l)) * 1.0) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(2.0 * l)) * 1.0) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (2.0 * l)) * 1.0) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 88.4

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

4. Applied rewrites 88.4%

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

5. Taylor expanded in K around 0

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

7. Applied rewrites 72.0%

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

8. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot 1 + U \]
9. Step-by-step derivation

10. Applied rewrites 54.3%

    \[\leadsto \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot 1 + U \]
11. Add Preprocessing

Alternative 18: 36.5% accurate, 330.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2025096 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))