Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.9%

Time: 5.5s

Alternatives: 16

Speedup: 1.2×

• 49.4% 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: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos \left(0.5 \cdot K\right) \cdot \sinh \ell\right) \cdot \left(J + J\right)\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (cos (* 0.5 K)) (sinh l)) (+ J J)))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 2e-169) (fma (* (cos (* -0.5 K)) (+ l l)) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (cos((0.5 * K)) * sinh(l)) * (J + J);
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 2e-169) {
		tmp = fma((cos((-0.5 * K)) * (l + l)), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(cos(Float64(0.5 * K)) * 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 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 2e-169)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(l + l)), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $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, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 2e-169], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * J + 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)))) < -inf.0 or 2.00000000000000004e-169 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.9%

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

   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

   6. Applied rewrites 65.0%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.00000000000000004e-169

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sinh-undef N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

Alternative 3: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Applied rewrites 99.9%

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

Alternative 4: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.99998:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.99998)
   (fma (* (cos (* -0.5 K)) (* (fma (* l l) 0.3333333333333333 2.0) l)) J U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.99998) {
		tmp = fma((cos((-0.5 * K)) * (fma((l * l), 0.3333333333333333, 2.0) * l)), J, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.99998)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)), J, U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.99998], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99997999999999998

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sinh-undef N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

   if 0.99997999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 5: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.25:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.25)
   (fma (* (cos (* -0.5 K)) (+ l l)) J U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.25) {
		tmp = fma((cos((-0.5 * K)) * (l + l)), J, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.25)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(l + l)), J, U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.25], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.25

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sinh-undef N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

   if 0.25 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 6: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.25:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.25)
   (fma (* (+ l l) J) (cos (* -0.5 K)) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.25) {
		tmp = fma(((l + l) * J), cos((-0.5 * K)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.25)
		tmp = fma(Float64(Float64(l + l) * J), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.25], N[(N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.25

   1. Initial program 86.1%

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

   2. Taylor expanded in 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 63.9

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

   4. Applied rewrites 63.9%

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

   if 0.25 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 7: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.6

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

   4. Applied rewrites 63.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 8: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \sinh \ell, J + J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* (fma (* K K) -0.125 1.0) (sinh l)) (+ J J) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((fma((K * K), -0.125, 1.0) * sinh(l)), (J + J), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * sinh(l)), Float64(J + J), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.6

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

   4. Applied rewrites 63.6%

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\sinh \ell \cdot 2\right), J, U\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 68.1%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 9: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot \left(\sinh \ell \cdot 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* (* (* K K) -0.125) (* (sinh l) 2.0)) J U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((((K * K) * -0.125) * (sinh(l) * 2.0)), J, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(Float64(Float64(K * K) * -0.125) * Float64(sinh(l) * 2.0)), J, U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.6

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

   4. Applied rewrites 63.6%

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 10: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma
    (fma -0.125 (* K K) 1.0)
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(fma(-0.125, (K * K), 1.0), ((fma((l * l), 0.3333333333333333, 2.0) * l) * J), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(fma(-0.125, Float64(K * K), 1.0), Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sinh-undef N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. cos-neg-rev N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 61.3

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

   11. Applied rewrites 61.3%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 11: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, \mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.125, J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (+ l l) (fma (* (* K K) J) -0.125 J) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((l + l), fma(((K * K) * J), -0.125, J), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(l + l), fma(Float64(Float64(K * K) * J), -0.125, J), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(l + l), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * J), $MachinePrecision] * -0.125 + J), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.1%

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

   2. Taylor expanded in 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 63.9

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

   4. Applied rewrites 63.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l* N/A

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

      8. cos-neg-rev N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 63.9%

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

   7. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 47.0

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

   9. Applied rewrites 47.0%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 12: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ t_1 := \left(J + J\right) \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))
        (t_1 (* (+ J J) (sinh l))))
   (if (<= t_0 (- INFINITY)) t_1 (if (<= t_0 1e+304) (fma (+ J J) l U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double t_1 = (J + J) * sinh(l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 1e+304) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	t_1 = Float64(Float64(J + J) * sinh(l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 1e+304)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 1e+304], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < -inf.0 or 9.9999999999999994e303 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in J around inf

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. sinh-undef N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sinh.f64 46.7

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

   7. Applied rewrites 46.7%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999994e303

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 54.1%

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

Alternative 13: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{J + J}{U} \cdot \ell, U, U\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* (/ (+ J J) U) l) U U)))
   (if (<= l -3.4e+106)
     t_0
     (if (<= l -850.0)
       (* (* l (fma -0.25 (* K K) 2.0)) J)
       (if (<= l 5e+33) (fma (+ J J) l U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((((J + J) / U) * l), U, U);
	double tmp;
	if (l <= -3.4e+106) {
		tmp = t_0;
	} else if (l <= -850.0) {
		tmp = (l * fma(-0.25, (K * K), 2.0)) * J;
	} else if (l <= 5e+33) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(Float64(Float64(J + J) / U) * l), U, U)
	tmp = 0.0
	if (l <= -3.4e+106)
		tmp = t_0;
	elseif (l <= -850.0)
		tmp = Float64(Float64(l * fma(-0.25, Float64(K * K), 2.0)) * J);
	elseif (l <= 5e+33)
		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[(N[(N[(J + J), $MachinePrecision] / U), $MachinePrecision] * l), $MachinePrecision] * U + U), $MachinePrecision]}, If[LessEqual[l, -3.4e+106], t$95$0, If[LessEqual[l, -850.0], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 5e+33], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.39999999999999994e106 or 4.99999999999999973e33 < l

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in U around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. rec-exp N/A

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

      7. associate-/l* N/A

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

      8. lower-/.f64 N/A

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

      9. sinh-undef N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. lift-sinh.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in l around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 55.1

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

   10. Applied rewrites 55.1%

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

   if -3.39999999999999994e106 < l < -850

   1. Initial program 86.1%

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

   2. Taylor expanded in 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      2. associate-+l+ N/A

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

      3. *-commutative N/A

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4} + \left(2 \cdot \left(J \cdot \ell\right) + U\right) \]

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left({K}^{2} \cdot \ell\right), \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, 2 \cdot \left(J \cdot \ell\right) + U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, 2 \cdot \left(\ell \cdot J\right) + U\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \left(2 \cdot \ell\right) \cdot J + U\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \mathsf{fma}\left(2 \cdot \ell, J, U\right)\right) \]

      15. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

      16. lift-+.f64 44.3

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, -0.25, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \color{blue}{-0.25}, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

   8. Taylor expanded in J around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right) \cdot J \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right) \cdot J \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell + 2 \cdot \ell\right) \cdot J \]

      4. distribute-rgt-out N/A

         \[\leadsto \left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right) \cdot J \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)\right) \cdot J \]

      7. pow2 N/A

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, K \cdot K, 2\right)\right) \cdot J \]

      8. lift-*.f64 25.2

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J \]

   10. Applied rewrites 25.2%

       \[\leadsto \left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J \]

   if -850 < l < 4.99999999999999973e33

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 54.1%

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

Alternative 14: 58.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -850:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* l (fma -0.25 (* K K) 2.0)) J)))
   (if (<= l -850.0) t_0 (if (<= l 1.1e+22) (fma (+ J J) l U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l * fma(-0.25, (K * K), 2.0)) * J;
	double tmp;
	if (l <= -850.0) {
		tmp = t_0;
	} else if (l <= 1.1e+22) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l * fma(-0.25, Float64(K * K), 2.0)) * J)
	tmp = 0.0
	if (l <= -850.0)
		tmp = t_0;
	elseif (l <= 1.1e+22)
		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[(l * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -850.0], t$95$0, If[LessEqual[l, 1.1e+22], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -850 or 1.1e22 < l

   1. Initial program 86.1%

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

   2. Taylor expanded in 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right) + U \]

      2. associate-+l+ N/A

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

      3. *-commutative N/A

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot \frac{-1}{4} + \left(2 \cdot \left(J \cdot \ell\right) + U\right) \]

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left({K}^{2} \cdot \ell\right), \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({K}^{2} \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, U + 2 \cdot \left(J \cdot \ell\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, 2 \cdot \left(J \cdot \ell\right) + U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, 2 \cdot \left(\ell \cdot J\right) + U\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \left(2 \cdot \ell\right) \cdot J + U\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \mathsf{fma}\left(2 \cdot \ell, J, U\right)\right) \]

      15. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \frac{-1}{4}, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

      16. lift-+.f64 44.3

          \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, -0.25, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \color{blue}{-0.25}, \mathsf{fma}\left(\ell + \ell, J, U\right)\right) \]

   8. Taylor expanded in J around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right) \cdot J \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left({K}^{2} \cdot \ell\right) + 2 \cdot \ell\right) \cdot J \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell + 2 \cdot \ell\right) \cdot J \]

      4. distribute-rgt-out N/A

         \[\leadsto \left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right) \cdot J \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)\right) \cdot J \]

      7. pow2 N/A

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{4}, K \cdot K, 2\right)\right) \cdot J \]

      8. lift-*.f64 25.2

         \[\leadsto \left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J \]

   10. Applied rewrites 25.2%

       \[\leadsto \left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J \]

   if -850 < l < 1.1e22

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sinh.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 54.1%

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

Alternative 15: 54.1% 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 86.1%

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

2. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. sinh-undef N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 N/A

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

   12. lower-sinh.f64 81.1

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in l around 0

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

7. Applied rewrites 54.1%

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

Alternative 16: 37.0% 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 86.1%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.0%

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

Reproduce

herbie shell --seed 2025129 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))