Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 100.0%

Time: 5.5s

Alternatives: 13

Speedup: 1.2×

• 49.7% 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: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(J + J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -0.0036:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.06:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (+ J J) (cos (* K 0.5))) (sinh l))))
   (if (<= l -0.0036)
     t_0
     (if (<= l 0.06) (fma (+ J J) (* (cos (* 0.5 K)) l) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J + J) * cos((K * 0.5))) * sinh(l);
	double tmp;
	if (l <= -0.0036) {
		tmp = t_0;
	} else if (l <= 0.06) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J + J) * cos(Float64(K * 0.5))) * sinh(l))
	tmp = 0.0
	if (l <= -0.0036)
		tmp = t_0;
	elseif (l <= 0.06)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -0.0036], t$95$0, If[LessEqual[l, 0.06], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0035999999999999999 or 0.059999999999999998 < l

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. sinh-undef-rev N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sinh.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if -0.0035999999999999999 < l < 0.059999999999999998

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-cos.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 64.3

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

   4. Applied rewrites 64.3%

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

Alternative 2: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

3. Applied rewrites 100.0%

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

Alternative 3: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq -0.106:\\ \;\;\;\;\left(\left(\frac{U}{J} + e^{\ell}\right) - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 2100000:\\ \;\;\;\;\mathsf{fma}\left(J + J, t\_0 \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 2\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))))
   (if (<= l -1.02e+138)
     (fma t_0 (* (* (* (* l l) J) 0.3333333333333333) l) U)
     (if (<= l -0.106)
       (* (- (+ (/ U J) (exp l)) (exp (- l))) J)
       (if (<= l 2100000.0)
         (fma (+ J J) (* t_0 l) U)
         (fma (* (* (sinh l) 2.0) J) (fma (* K K) -0.125 1.0) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double tmp;
	if (l <= -1.02e+138) {
		tmp = fma(t_0, ((((l * l) * J) * 0.3333333333333333) * l), U);
	} else if (l <= -0.106) {
		tmp = (((U / J) + exp(l)) - exp(-l)) * J;
	} else if (l <= 2100000.0) {
		tmp = fma((J + J), (t_0 * l), U);
	} else {
		tmp = fma(((sinh(l) * 2.0) * J), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (l <= -1.02e+138)
		tmp = fma(t_0, Float64(Float64(Float64(Float64(l * l) * J) * 0.3333333333333333) * l), U);
	elseif (l <= -0.106)
		tmp = Float64(Float64(Float64(Float64(U / J) + exp(l)) - exp(Float64(-l))) * J);
	elseif (l <= 2100000.0)
		tmp = fma(Float64(J + J), Float64(t_0 * l), U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 2.0) * J), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.02e+138], N[(t$95$0 * N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, -0.106], N[(N[(N[(N[(U / J), $MachinePrecision] + N[Exp[l], $MachinePrecision]), $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 2100000.0], N[(N[(J + J), $MachinePrecision] * N[(t$95$0 * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.02e138

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 71.4

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

   7. Applied rewrites 71.4%

      \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

   9. Applied rewrites 71.4%

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

   if -1.02e138 < l < -0.105999999999999997

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 56.4

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

   9. Applied rewrites 56.4%

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

   if -0.105999999999999997 < l < 2.1e6

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-cos.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 64.3

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

   4. Applied rewrites 64.3%

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

   if 2.1e6 < l

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 69.4%

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

Alternative 4: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.675:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.675)
   (fma (* (cos (* 0.5 K)) J) (* (fma (* l l) 0.3333333333333333 2.0) l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.675) {
		tmp = fma((cos((0.5 * K)) * J), (fma((l * l), 0.3333333333333333, 2.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.675)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.675], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.67500000000000004

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.3

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

   6. Applied rewrites 88.3%

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

   if 0.67500000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.9

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

   4. Applied rewrites 80.9%

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

Alternative 5: 88.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.44:\\ \;\;\;\;\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.09:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.44)
     (+
      (* (* (* (* (* l l) l) J) 0.3333333333333333) (fma (* K K) -0.125 1.0))
      U)
     (if (<= t_0 0.09)
       (fma (+ J J) (* (cos (* 0.5 K)) l) 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 <= -0.44) {
		tmp = (((((l * l) * l) * J) * 0.3333333333333333) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= 0.09) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), 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 <= -0.44)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.3333333333333333) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= 0.09)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), 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, -0.44], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.09], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.440000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 61.2

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

   10. Applied rewrites 61.2%

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

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow3 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 57.0

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

   13. Applied rewrites 57.0%

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

   if -0.440000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.089999999999999997

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-cos.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 64.3

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

   4. Applied rewrites 64.3%

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

   if 0.089999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.9

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

   4. Applied rewrites 80.9%

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

Alternative 6: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.16:\\ \;\;\;\;\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.16)
   (+
    (* (* (* (* (* l l) l) J) 0.3333333333333333) (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)) <= -0.16) {
		tmp = (((((l * l) * l) * J) * 0.3333333333333333) * 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)) <= -0.16)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.3333333333333333) * 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], -0.16], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.160000000000000003

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 61.2

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

   10. Applied rewrites 61.2%

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

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow3 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 57.0

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

   13. Applied rewrites 57.0%

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

   if -0.160000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.9

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

   4. Applied rewrites 80.9%

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

Alternative 7: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\left(\left(J + J\right) \cdot \ell\right) \cdot \left(\sqrt{\left(K \cdot K\right) \cdot \left(K \cdot K\right)} \cdot -0.125\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ (* (* (+ J J) l) (* (sqrt (* (* K K) (* K K))) -0.125)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = (((J + J) * l) * (sqrt(((K * K) * (K * K))) * -0.125)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(Float64(Float64(Float64(J + J) * l) * Float64(sqrt(Float64(Float64(K * K) * Float64(K * K))) * -0.125)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[Sqrt[N[(N[(K * K), $MachinePrecision] * N[(K * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125), $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))) < -0.10000000000000001

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.0

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

   10. Applied rewrites 35.0%

       \[\leadsto \left(\left(J + J\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{-0.125}\right) + U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. fabs-pow2-rev N/A

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

       4. rem-sqrt-square-rev N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 33.0

           \[\leadsto \left(\left(J + J\right) \cdot \ell\right) \cdot \left(\sqrt{\left(K \cdot K\right) \cdot \left(K \cdot K\right)} \cdot -0.125\right) + U \]

   12. Applied rewrites 33.0%

       \[\leadsto \left(\left(J + J\right) \cdot \ell\right) \cdot \left(\sqrt{\left(K \cdot K\right) \cdot \left(K \cdot K\right)} \cdot -0.125\right) + U \]

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.9

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

   4. Applied rewrites 80.9%

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

Alternative 8: 85.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \left(K \cdot K\right) \cdot -0.125, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (fma (* (+ J J) l) (* (* K K) -0.125) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = fma(((J + J) * l), ((K * K) * -0.125), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = fma(Float64(Float64(J + J) * l), Float64(Float64(K * K) * -0.125), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.0

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

   10. Applied rewrites 35.0%

       \[\leadsto \left(\left(J + J\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{-0.125}\right) + U \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lower-fma.f64 35.0

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

   12. Applied rewrites 35.0%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 80.9

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

   4. Applied rewrites 80.9%

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

Alternative 9: 80.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \ell \cdot \left(J + J\right)\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 118:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot J, \ell \cdot \ell, J + J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (sinh l) (+ J J))))
   (if (<= l -3.3e-13)
     t_0
     (if (<= l 118.0)
       (fma (fma (* 0.3333333333333333 J) (* l l) (+ J J)) l U)
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = sinh(l) * (J + J);
	double tmp;
	if (l <= -3.3e-13) {
		tmp = t_0;
	} else if (l <= 118.0) {
		tmp = fma(fma((0.3333333333333333 * J), (l * l), (J + J)), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(sinh(l) * Float64(J + J))
	tmp = 0.0
	if (l <= -3.3e-13)
		tmp = t_0;
	elseif (l <= 118.0)
		tmp = fma(fma(Float64(0.3333333333333333 * J), Float64(l * l), Float64(J + J)), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.3e-13], t$95$0, If[LessEqual[l, 118.0], N[(N[(N[(0.3333333333333333 * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + N[(J + J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.3000000000000001e-13 or 118 < l

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. sinh-undef-rev N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if -3.3000000000000001e-13 < l < 118

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. count-2-rev N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied rewrites 69.8%

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

Alternative 10: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \ell \cdot \left(J + J\right)\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (sinh l) (+ J J)))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 -2e-158) t_0 (if (<= t_1 2e+281) (fma (+ J J) l U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = sinh(l) * (J + J);
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -2e-158) {
		tmp = t_0;
	} else if (t_1 <= 2e+281) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(sinh(l) * Float64(J + J))
	t_1 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= -2e-158)
		tmp = t_0;
	elseif (t_1 <= 2e+281)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-158], t$95$0, If[LessEqual[t$95$1, 2e+281], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -2.00000000000000013e-158 or 2.0000000000000001e281 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. sinh-undef-rev N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if -2.00000000000000013e-158 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.0000000000000001e281

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.7

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

   9. Applied rewrites 54.7%

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

Alternative 11: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \left(K \cdot K\right) \cdot -0.125, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (fma (* (+ J J) l) (* (* K K) -0.125) U)
   (fma (+ J J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = fma(((J + J) * l), ((K * K) * -0.125), U);
	} else {
		tmp = fma((J + J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = fma(Float64(Float64(J + J) * l), Float64(Float64(K * K) * -0.125), U);
	else
		tmp = fma(Float64(J + J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.0

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

   10. Applied rewrites 35.0%

       \[\leadsto \left(\left(J + J\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{-0.125}\right) + U \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lower-fma.f64 35.0

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

   12. Applied rewrites 35.0%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.7

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

   9. Applied rewrites 54.7%

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

Alternative 12: 54.7% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. rec-exp N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. sinh-undef-rev N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sinh.f64 80.9

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

6. Applied rewrites 80.9%

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

7. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-+.f64 54.7

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

9. Applied rewrites 54.7%

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

Alternative 13: 36.9% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.9%

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

Reproduce

herbie shell --seed 2025134 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))