Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.9% → 99.9%

Time: 4.7s

Alternatives: 14

Speedup: 1.2×

• 50.8% 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: 86.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \left(\left(J + J\right) \cdot t\_0\right) \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.0125:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\ell + \ell\right) \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (* (* (+ J J) t_0) (sinh l))))
   (if (<= l -0.02) t_1 (if (<= l 0.0125) (fma t_0 (* (+ l l) J) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = ((J + J) * t_0) * sinh(l);
	double tmp;
	if (l <= -0.02) {
		tmp = t_1;
	} else if (l <= 0.0125) {
		tmp = fma(t_0, ((l + l) * J), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(Float64(Float64(J + J) * t_0) * sinh(l))
	tmp = 0.0
	if (l <= -0.02)
		tmp = t_1;
	elseif (l <= 0.0125)
		tmp = fma(t_0, Float64(Float64(l + l) * J), U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -0.02], t$95$1, If[LessEqual[l, 0.0125], N[(t$95$0 * N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0200000000000000004 or 0.012500000000000001 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sinh-undef-rev N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sinh.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -0.0200000000000000004 < l < 0.012500000000000001

   1. Initial program 73.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

      12. sinh-undef-rev 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in l around 0

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

       1. count-2-rev N/A

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

       2. lift-+.f64 99.6

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

   12. Applied rewrites 99.6%

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

Alternative 3: 88.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5e-69)
   (fma (* 2.0 (sinh l)) J U)
   (fma (cos (* 0.5 K)) (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5e-69) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = fma(cos((0.5 * K)), ((fma((l * l), 0.3333333333333333, 2.0) * l) * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5e-69)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = fma(cos(Float64(0.5 * K)), Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5e-69], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 5.00000000000000033e-69

   1. Initial program 86.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 86.1

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

   4. Applied rewrites 86.1%

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

   if 5.00000000000000033e-69 < K

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 87.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 87.7

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

      12. sinh-undef-rev 87.7

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 87.7

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

   9. Applied rewrites 87.7%

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

Alternative 4: 87.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5e-69)
   (fma (* 2.0 (sinh l)) J U)
   (fma (* (cos (* 0.5 K)) (* (fma (* l l) 0.3333333333333333 2.0) l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5e-69) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = fma((cos((0.5 * K)) * (fma((l * l), 0.3333333333333333, 2.0) * l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5e-69)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5e-69], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if K < 5.00000000000000033e-69

   1. Initial program 86.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 86.1

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

   4. Applied rewrites 86.1%

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

   if 5.00000000000000033e-69 < K

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 87.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 87.7

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

      12. sinh-undef-rev 87.7

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 87.7

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

   9. Applied rewrites 87.7%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 87.7

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

       6. sinh-undef-rev 87.7

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

   11. Applied rewrites 87.7%

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

Alternative 5: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := 2 \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, J, U\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \left(\ell + \ell\right) \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J, t\_1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* 2.0 (sinh l))))
   (if (<= t_0 -2e+45)
     (fma t_1 J U)
     (if (<= t_0 5e+202)
       (fma (cos (* 0.5 K)) (* (+ l l) J) U)
       (fma (* (fma (* K K) -0.125 1.0) J) t_1 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = 2.0 * sinh(l);
	double tmp;
	if (t_0 <= -2e+45) {
		tmp = fma(t_1, J, U);
	} else if (t_0 <= 5e+202) {
		tmp = fma(cos((0.5 * K)), ((l + l) * J), U);
	} else {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), t_1, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (t_0 <= -2e+45)
		tmp = fma(t_1, J, U);
	elseif (t_0 <= 5e+202)
		tmp = fma(cos(Float64(0.5 * K)), Float64(Float64(l + l) * J), U);
	else
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), t_1, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+45], N[(t$95$1 * J + U), $MachinePrecision], If[LessEqual[t$95$0, 5e+202], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * t$95$1 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.9999999999999999e45

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if -1.9999999999999999e45 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 4.9999999999999999e202

   1. Initial program 73.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 99.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.6

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

      12. sinh-undef-rev 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in l around 0

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

       1. count-2-rev N/A

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

       2. lift-+.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 4.9999999999999999e202 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.7%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 6: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := 2 \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, J, U\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J, t\_1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* 2.0 (sinh l))))
   (if (<= t_0 -2e+45)
     (fma t_1 J U)
     (if (<= t_0 5e+202)
       (fma (* (+ J J) (cos (* 0.5 K))) l U)
       (fma (* (fma (* K K) -0.125 1.0) J) t_1 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = 2.0 * sinh(l);
	double tmp;
	if (t_0 <= -2e+45) {
		tmp = fma(t_1, J, U);
	} else if (t_0 <= 5e+202) {
		tmp = fma(((J + J) * cos((0.5 * K))), l, U);
	} else {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), t_1, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (t_0 <= -2e+45)
		tmp = fma(t_1, J, U);
	elseif (t_0 <= 5e+202)
		tmp = fma(Float64(Float64(J + J) * cos(Float64(0.5 * K))), l, U);
	else
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), t_1, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+45], N[(t$95$1 * J + U), $MachinePrecision], If[LessEqual[t$95$0, 5e+202], N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * t$95$1 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.9999999999999999e45

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if -1.9999999999999999e45 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 4.9999999999999999e202

   1. Initial program 73.3%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 4.9999999999999999e202 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.7%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 7: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.8%

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

   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 68.7

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

   4. Applied rewrites 68.7%

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

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

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 94.6

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

   4. Applied rewrites 94.6%

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

Alternative 8: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J, t\_0, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma (* (fma (* K K) -0.125 1.0) J) t_0 U)
     (fma t_0 J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), t_0, U);
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), t_0, U);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * t$95$0 + U), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.8%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.7

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

   7. Applied rewrites 63.7%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 9: 86.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.8%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 61.4

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

   4. Applied rewrites 61.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.6

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

   7. Applied rewrites 55.6%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 10: 63.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5 - 1\right)\right) \cdot 1 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)))
   (if (<= l -1.1e-23)
     t_0
     (if (<= l 0.66)
       (fma (+ J J) l U)
       (if (<= l 1.05e+45) t_0 (+ (* (* J (- (* (* l l) 0.5) 1.0)) 1.0) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	double tmp;
	if (l <= -1.1e-23) {
		tmp = t_0;
	} else if (l <= 0.66) {
		tmp = fma((J + J), l, U);
	} else if (l <= 1.05e+45) {
		tmp = t_0;
	} else {
		tmp = ((J * (((l * l) * 0.5) - 1.0)) * 1.0) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U)
	tmp = 0.0
	if (l <= -1.1e-23)
		tmp = t_0;
	elseif (l <= 0.66)
		tmp = fma(Float64(J + J), l, U);
	elseif (l <= 1.05e+45)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(l * l) * 0.5) - 1.0)) * 1.0) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.1e-23], t$95$0, If[LessEqual[l, 0.66], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[l, 1.05e+45], t$95$0, N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e-23 or 0.660000000000000031 < l < 1.04999999999999997e45

   1. Initial program 98.4%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 29.4

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.2

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

   7. Applied rewrites 34.2%

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

   if -1.1e-23 < l < 0.660000000000000031

   1. Initial program 73.2%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 87.2%

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

   if 1.04999999999999997e45 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 56.0

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1\right)\right) \cdot 1 + U \]

   10. Applied rewrites 56.0%

       \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right)} - 1\right)\right) \cdot 1 + U \]

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 56.0

          \[\leadsto \left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5 - 1\right)\right) \cdot 1 + U \]

   13. Applied rewrites 56.0%

       \[\leadsto \left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{0.5} - 1\right)\right) \cdot 1 + U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot 0.5, \ell, 1\right) - 1\right)\right) \cdot 1 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)))
   (if (<= l -1.1e-23)
     t_0
     (if (<= l 0.66)
       (fma (+ J J) l U)
       (if (<= l 1.05e+45)
         t_0
         (+ (* (* J (- (fma (* l 0.5) l 1.0) 1.0)) 1.0) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	double tmp;
	if (l <= -1.1e-23) {
		tmp = t_0;
	} else if (l <= 0.66) {
		tmp = fma((J + J), l, U);
	} else if (l <= 1.05e+45) {
		tmp = t_0;
	} else {
		tmp = ((J * (fma((l * 0.5), l, 1.0) - 1.0)) * 1.0) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U)
	tmp = 0.0
	if (l <= -1.1e-23)
		tmp = t_0;
	elseif (l <= 0.66)
		tmp = fma(Float64(J + J), l, U);
	elseif (l <= 1.05e+45)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * 0.5), l, 1.0) - 1.0)) * 1.0) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.1e-23], t$95$0, If[LessEqual[l, 0.66], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[l, 1.05e+45], t$95$0, N[(N[(N[(J * N[(N[(N[(l * 0.5), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e-23 or 0.660000000000000031 < l < 1.04999999999999997e45

   1. Initial program 98.4%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 29.4

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.2

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

   7. Applied rewrites 34.2%

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

   if -1.1e-23 < l < 0.660000000000000031

   1. Initial program 73.2%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 87.2%

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

   if 1.04999999999999997e45 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 56.0

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1\right)\right) \cdot 1 + U \]

   10. Applied rewrites 56.0%

       \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right)} - 1\right)\right) \cdot 1 + U \]

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 56.0

          \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot 0.5, \ell, 1\right) - 1\right)\right) \cdot 1 + U \]

   13. Applied rewrites 56.0%

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot 0.5, \ell, 1\right) - 1\right)\right) \cdot 1 + U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8000000:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5 - 1\right)\right) \cdot 1 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 8000000.0)
   (fma (+ J J) l U)
   (+ (* (* J (- (* (* l l) 0.5) 1.0)) 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 8000000.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = ((J * (((l * l) * 0.5) - 1.0)) * 1.0) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 8000000.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(Float64(Float64(l * l) * 0.5) - 1.0)) * 1.0) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 8000000.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8e6

   1. Initial program 82.6%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 63.6%

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

   if 8e6 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.1

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1\right)\right) \cdot 1 + U \]

   10. Applied rewrites 50.1%

       \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right)} - 1\right)\right) \cdot 1 + U \]

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.1

          \[\leadsto \left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5 - 1\right)\right) \cdot 1 + U \]

   13. Applied rewrites 50.1%

       \[\leadsto \left(J \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{0.5} - 1\right)\right) \cdot 1 + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.5% 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 86.9%

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

2. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 63.0

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in K around 0

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

7. Applied rewrites 53.5%

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

Alternative 14: 36.2% 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 86.9%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.2%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))