Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.9%

Time: 7.1s

Alternatives: 19

Speedup: 1.5×

• 49.9% 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: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. sinh-undef N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sinh.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in K around 0

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

   1. lift-*.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.7138:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.7138)
     (+
      (*
       (*
        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 <= 0.7138) {
		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 <= 0.7138)
		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, 0.7138], 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))) < 0.71379999999999999

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   if 0.71379999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 3: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.7138:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.7138)
     (+
      (*
       (*
        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 <= 0.7138) {
		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 <= 0.7138)
		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, 0.7138], 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))) < 0.71379999999999999

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-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 91.8

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

   5. Applied rewrites 91.8%

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

   if 0.71379999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 4: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.191:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.191)
   (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)) <= 0.191) {
		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)) <= 0.191)
		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], 0.191], 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))) < 0.191

   1. Initial program 84.7%

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

   if 0.191 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.191:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (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)) <= 0.055) {
		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)) <= 0.055)
		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], 0.055], 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))) < 0.0550000000000000003

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 71.6

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

   5. Applied rewrites 71.6%

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

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 6: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;\ell \leq -74 \lor \neg \left(\ell \leq 29\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, t\_0, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))))
   (if (or (<= l -74.0) (not (<= l 29.0)))
     (* (* t_0 J) (* 2.0 (sinh l)))
     (fma (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double tmp;
	if ((l <= -74.0) || !(l <= 29.0)) {
		tmp = (t_0 * J) * (2.0 * sinh(l));
	} else {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), t_0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	tmp = 0.0
	if ((l <= -74.0) || !(l <= 29.0))
		tmp = Float64(Float64(t_0 * J) * Float64(2.0 * sinh(l)));
	else
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), t_0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[l, -74.0], N[Not[LessEqual[l, 29.0]], $MachinePrecision]], N[(N[(t$95$0 * J), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -74 or 29 < 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. Add Preprocessing

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -74 < l < 29

   1. Initial program 75.3%

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -74 \lor \neg \left(\ell \leq 29\right):\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    (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)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), 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)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), 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], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $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))) < -2e-3

   1. Initial program 86.6%

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.2

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 56.3

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

   13. Applied rewrites 56.3%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 9: 83.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* l l)
        0.3333333333333333)
       (* l l)
       2.0)
      l)
     J)
    1.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l) * J), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * J), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.2

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 56.3

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

   13. Applied rewrites 56.3%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 90.7%

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \color{blue}{1} + U \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.7

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

   10. Applied rewrites 90.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (fma (fma l (* 0.016666666666666666 l) 0.3333333333333333) (* l l) 2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma(fma(l, (0.016666666666666666 * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(fma(l, Float64(0.016666666666666666 * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * N[(0.016666666666666666 * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.2

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 56.3

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

   13. Applied rewrites 56.3%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 89.2

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

   8. Applied rewrites 89.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 89.2

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

   10. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (* l J) (fma (* K K) -0.125 1.0)) 2.0 U)
   (fma
    (*
     (fma (fma l (* 0.016666666666666666 l) 0.3333333333333333) (* l l) 2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((l * J) * fma((K * K), -0.125, 1.0)), 2.0, U);
	} else {
		tmp = fma((fma(fma(l, (0.016666666666666666 * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(l * J) * fma(Float64(K * K), -0.125, 1.0)), 2.0, U);
	else
		tmp = fma(Float64(fma(fma(l, Float64(0.016666666666666666 * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(N[(l * N[(0.016666666666666666 * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 89.2

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

   8. Applied rewrites 89.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 89.2

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

   10. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.016666666666666666 \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666, \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (* l J) (fma (* K K) -0.125 1.0)) 2.0 U)
   (fma (* (fma (* (* l l) 0.016666666666666666) (* l l) 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((l * J) * fma((K * K), -0.125, 1.0)), 2.0, U);
	} else {
		tmp = fma((fma(((l * l) * 0.016666666666666666), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(l * J) * fma(Float64(K * K), -0.125, 1.0)), 2.0, U);
	else
		tmp = fma(Float64(fma(Float64(Float64(l * l) * 0.016666666666666666), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 89.2

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 89.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666, \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]

   11. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666, \ell \cdot \ell, 2\right) \cdot \ell, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666, \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* (* l J) (fma (* K K) -0.125 1.0)) 2.0 U)
   (fma (* (fma l (* l 0.3333333333333333) 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((l * J) * fma((K * K), -0.125, 1.0)), 2.0, U);
	} else {
		tmp = fma((fma(l, (l * 0.3333333333333333), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(l * J) * fma(Float64(K * K), -0.125, 1.0)), 2.0, U);
	else
		tmp = fma(Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 85.3

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

   8. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.3

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

   10. Applied rewrites 85.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\left(\left(J \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.82)
   (* (* (* J l) (fma (* K K) -0.125 1.0)) 2.0)
   (fma (* (fma l (* l 0.3333333333333333) 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.82) {
		tmp = ((J * l) * fma((K * K), -0.125, 1.0)) * 2.0;
	} else {
		tmp = fma((fma(l, (l * 0.3333333333333333), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.82)
		tmp = Float64(Float64(Float64(J * l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0);
	else
		tmp = fma(Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.82], N[(N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 48.4

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 43.2

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

   11. Applied rewrites 43.2%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.6%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 79.2

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

   10. Applied rewrites 79.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\left(\left(J \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.26 \cdot 10^{-45}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.6e-7) (not (<= l 1.26e-45)))
   (fma (* (* (* l l) 0.3333333333333333) l) J U)
   (fma (* J l) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.6e-7) || !(l <= 1.26e-45)) {
		tmp = fma((((l * l) * 0.3333333333333333) * l), J, U);
	} else {
		tmp = fma((J * l), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.6e-7) || !(l <= 1.26e-45))
		tmp = fma(Float64(Float64(Float64(l * l) * 0.3333333333333333) * l), J, U);
	else
		tmp = fma(Float64(J * l), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.6e-7], N[Not[LessEqual[l, 1.26e-45]], $MachinePrecision]], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J * l), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.6e-7 or 1.26e-45 < l

   1. Initial program 99.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. 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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 63.4

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

   11. Applied rewrites 63.4%

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

   if -1.6e-7 < l < 1.26e-45

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 88.8

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

   8. Applied rewrites 88.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.26 \cdot 10^{-45}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.6% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-56} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+17}\right):\\ \;\;\;\;\left(J \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.5e-56) (not (<= l 1.9e+17))) (* (* J l) 2.0) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.5e-56) || !(l <= 1.9e+17)) {
		tmp = (J * l) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6.5d-56)) .or. (.not. (l <= 1.9d+17))) then
        tmp = (j * l) * 2.0d0
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.5e-56) || !(l <= 1.9e+17)) {
		tmp = (J * l) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.5e-56) or not (l <= 1.9e+17):
		tmp = (J * l) * 2.0
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.5e-56) || !(l <= 1.9e+17))
		tmp = Float64(Float64(J * l) * 2.0);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.5e-56) || ~((l <= 1.9e+17)))
		tmp = (J * l) * 2.0;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.5e-56], N[Not[LessEqual[l, 1.9e+17]], $MachinePrecision]], N[(N[(J * l), $MachinePrecision] * 2.0), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.4999999999999997e-56 or 1.9e17 < l

   1. Initial program 93.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. 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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 38.5

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in K around 0

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

       1. lift-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if -6.4999999999999997e-56 < l < 1.9e17

   1. Initial program 81.0%

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

   3. Taylor expanded in J around 0

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

   5. Applied rewrites 77.5%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-56} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+17}\right):\\ \;\;\;\;\left(J \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.5% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right) \end{array} \]

FPCore

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

C

double code(double J, double l, double K, double U) {
	return fma((fma(l, (l * 0.3333333333333333), 2.0) * l), J, U);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. 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 83.8

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

5. Applied rewrites 83.8%

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

6. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 75.1

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

8. Applied rewrites 75.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 75.1

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

10. Applied rewrites 75.1%

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

11. Final simplification 75.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \ell, J, U\right) \]
12. Add Preprocessing

Alternative 18: 54.4% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. 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 66.4

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

5. Applied rewrites 66.4%

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

6. Taylor expanded in K around 0

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

   1. lower-*.f64 57.3

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

8. Applied rewrites 57.3%

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

Alternative 19: 36.8% accurate, 330.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in J around 0

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

5. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2025056 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))