Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.7% → 99.9%

Time: 5.6s

Alternatives: 17

Speedup: 1.2×

• 48.5% 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: 85.7% 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 85.7%

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

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 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \left(\left(J + J\right) \cdot t\_1\right) \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, t\_1 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* (* (+ J J) t_1) (sinh l))))
   (if (<= t_0 -4e+30)
     t_2
     (if (<= t_0 1e-158) (fma (+ l l) (* t_1 J) U) t_2))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double t_1 = cos((0.5 * K));
	double t_2 = ((J + J) * t_1) * sinh(l);
	double tmp;
	if (t_0 <= -4e+30) {
		tmp = t_2;
	} else if (t_0 <= 1e-158) {
		tmp = fma((l + l), (t_1 * J), U);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(Float64(Float64(J + J) * t_1) * sinh(l))
	tmp = 0.0
	if (t_0 <= -4e+30)
		tmp = t_2;
	elseif (t_0 <= 1e-158)
		tmp = fma(Float64(l + l), Float64(t_1 * J), U);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(J + J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+30], t$95$2, If[LessEqual[t$95$0, 1e-158], N[(N[(l + l), $MachinePrecision] * N[(t$95$1 * J), $MachinePrecision] + U), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -4.0000000000000001e30 or 1.00000000000000006e-158 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.2%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 98.7

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

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sinh.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. rec-exp N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. rec-exp N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sinh-undef-rev N/A

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

   6. Applied rewrites 98.6%

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

   if -4.0000000000000001e30 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 1.00000000000000006e-158

   1. Initial program 72.1%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

   7. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \left(t\_0 \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -82:\\ \;\;\;\;\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J\\ \mathbf{elif}\;\ell \leq 110:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, t\_0 \cdot J, U\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+82}:\\ \;\;\;\;\sinh \ell \cdot \left(J + J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (* (* t_0 (* (fma (* l l) 0.3333333333333333 2.0) l)) J)))
   (if (<= l -1.15e+86)
     t_1
     (if (<= l -82.0)
       (* (* (fma (* K K) -0.125 1.0) (* 2.0 (sinh l))) J)
       (if (<= l 110.0)
         (fma (+ l l) (* t_0 J) U)
         (if (<= l 1.02e+82) (* (sinh l) (+ J J)) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = (t_0 * (fma((l * l), 0.3333333333333333, 2.0) * l)) * J;
	double tmp;
	if (l <= -1.15e+86) {
		tmp = t_1;
	} else if (l <= -82.0) {
		tmp = (fma((K * K), -0.125, 1.0) * (2.0 * sinh(l))) * J;
	} else if (l <= 110.0) {
		tmp = fma((l + l), (t_0 * J), U);
	} else if (l <= 1.02e+82) {
		tmp = sinh(l) * (J + J);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(Float64(t_0 * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * J)
	tmp = 0.0
	if (l <= -1.15e+86)
		tmp = t_1;
	elseif (l <= -82.0)
		tmp = Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(2.0 * sinh(l))) * J);
	elseif (l <= 110.0)
		tmp = fma(Float64(l + l), Float64(t_0 * J), U);
	elseif (l <= 1.02e+82)
		tmp = Float64(sinh(l) * Float64(J + J));
	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[(t$95$0 * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1.15e+86], t$95$1, If[LessEqual[l, -82.0], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 110.0], N[(N[(l + l), $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.02e+82], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.14999999999999995e86 or 1.0200000000000001e82 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if -1.14999999999999995e86 < l < -82

   1. Initial program 99.8%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -82 < l < 110

   1. Initial program 72.2%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

   7. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if 110 < l < 1.0200000000000001e82

   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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   6. 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-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 77.0

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

   10. Applied rewrites 77.0%

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

Alternative 4: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.05)
   (fma (* (cos (* 0.5 K)) J) (* (fma (* l l) 0.3333333333333333 2.0) l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.05) {
		tmp = fma((cos((0.5 * K)) * J), (fma((l * l), 0.3333333333333333, 2.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.05)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.05], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.4%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

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

   1. Initial program 85.7%

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

   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 95.9

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

   4. Applied rewrites 95.9%

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

Alternative 5: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8)
     (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
     (if (<= t_0 -0.01)
       (fma (+ J J) (* (cos (* 0.5 K)) l) U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= -0.01) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], 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], If[LessEqual[t$95$0, -0.01], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 85.5%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 65.9

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

   4. Applied rewrites 65.9%

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

   if -0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 84.8%

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

   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 66.3

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

   4. Applied rewrites 66.3%

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

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

   1. Initial program 85.8%

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

   2. Taylor expanded in 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 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell + \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8)
     (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
     (if (<= t_0 -0.01)
       (fma (cos (* 0.5 K)) (* J (+ l l)) U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= -0.01) {
		tmp = fma(cos((0.5 * K)), (J * (l + l)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= -0.01)
		tmp = fma(cos(Float64(0.5 * K)), Float64(J * Float64(l + l)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], 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], If[LessEqual[t$95$0, -0.01], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 85.5%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 65.9

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

   4. Applied rewrites 65.9%

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

   if -0.80000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 84.8%

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

   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 l around 0

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

   7. Applied rewrites 66.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 66.3

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 66.3

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

   9. Applied rewrites 66.3%

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

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

   1. Initial program 85.8%

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

   2. Taylor expanded in 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 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ (* (* 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)) <= -0.01) {
		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)) <= -0.01)
		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], -0.01], 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))) < -0.0100000000000000002

   1. Initial program 85.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

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

   1. Initial program 85.8%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 8: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot J, \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.015)
   (fma (* (* (* K K) -0.125) J) (* (fma (* l l) 0.3333333333333333 2.0) l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.015) {
		tmp = fma((((K * K) * -0.125) * J), (fma((l * l), 0.3333333333333333, 2.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.015)
		tmp = fma(Float64(Float64(Float64(K * K) * -0.125) * J), Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.015], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] * J), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 85.2%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.8

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

   10. Applied rewrites 57.8%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 57.8

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

   13. Applied rewrites 57.8%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   2. Taylor expanded in 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 95.7

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

   4. Applied rewrites 95.7%

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

Alternative 9: 85.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;\left(\left(J + J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.015)
   (+ (* (* (+ J J) l) (fma (* K K) -0.125 1.0)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.015) {
		tmp = (((J + J) * l) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.015)
		tmp = Float64(Float64(Float64(Float64(J + J) * l) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.015], N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 85.2%

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

   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 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 52.1

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

   7. Applied rewrites 52.1%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   2. Taylor expanded in 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 95.7

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

   4. Applied rewrites 95.7%

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

Alternative 10: 78.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \ell \cdot \left(J + J\right)\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+216}:\\ \;\;\;\;\left(\left(\ell + \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot J\\ \mathbf{elif}\;\ell \leq -68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 110:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (sinh l) (+ J J))))
   (if (<= l -2.25e+216)
     (* (* (+ l l) (fma (* K K) -0.125 1.0)) J)
     (if (<= l -68.0) t_0 (if (<= l 110.0) (fma (+ l l) J U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = sinh(l) * (J + J);
	double tmp;
	if (l <= -2.25e+216) {
		tmp = ((l + l) * fma((K * K), -0.125, 1.0)) * J;
	} else if (l <= -68.0) {
		tmp = t_0;
	} else if (l <= 110.0) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(sinh(l) * Float64(J + J))
	tmp = 0.0
	if (l <= -2.25e+216)
		tmp = Float64(Float64(Float64(l + l) * fma(Float64(K * K), -0.125, 1.0)) * J);
	elseif (l <= -68.0)
		tmp = t_0;
	elseif (l <= 110.0)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.25e+216], N[(N[(N[(l + l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, -68.0], t$95$0, If[LessEqual[l, 110.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.25000000000000012e216

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 45.3

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

   7. Applied rewrites 45.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.9

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

   10. Applied rewrites 41.9%

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

   if -2.25000000000000012e216 < l < -68 or 110 < 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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   6. 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-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 75.1

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

   7. Applied rewrites 75.1%

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

   8. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 75.0

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

   10. Applied rewrites 75.0%

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

   if -68 < l < 110

   1. Initial program 72.2%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

   7. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in K around 0

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

   12. Applied rewrites 86.0%

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

Alternative 11: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;\left(\left(\ell + \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.015)
   (* (* (+ l l) (fma (* K K) -0.125 1.0)) J)
   (fma J (* (fma (* l l) 0.3333333333333333 2.0) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.015) {
		tmp = ((l + l) * fma((K * K), -0.125, 1.0)) * J;
	} else {
		tmp = fma(J, (fma((l * l), 0.3333333333333333, 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.015)
		tmp = Float64(Float64(Float64(l + l) * fma(Float64(K * K), -0.125, 1.0)) * J);
	else
		tmp = fma(J, Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.015], N[(N[(N[(l + l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 85.2%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 65.3

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   2. Taylor expanded in 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 l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 84.4%

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

Alternative 12: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot 0.16666666666666666\right) \cdot 2\right) \cdot J\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* l l) l) 0.16666666666666666) 2.0) J))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 -4e+30) t_0 (if (<= t_1 2e+183) (fma (+ l l) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((((l * l) * l) * 0.16666666666666666) * 2.0) * J;
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -4e+30) {
		tmp = t_0;
	} else if (t_1 <= 2e+183) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(Float64(Float64(l * l) * l) * 0.16666666666666666) * 2.0) * J)
	t_1 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= -4e+30)
		tmp = t_0;
	elseif (t_1 <= 2e+183)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+30], t$95$0, If[LessEqual[t$95$1, 2e+183], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -4.0000000000000001e30 or 1.99999999999999989e183 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in K around 0

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

      1. rec-exp N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sinh.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 57.6

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

   10. Applied rewrites 57.6%

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

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

          \[\leadsto \left(\left({\ell}^{3} \cdot \frac{1}{6}\right) \cdot 2\right) \cdot J \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({\ell}^{3} \cdot \frac{1}{6}\right) \cdot 2\right) \cdot J \]

       3. unpow3 N/A

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

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 57.0

          \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot 0.16666666666666666\right) \cdot 2\right) \cdot J \]

   13. Applied rewrites 57.0%

       \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot 0.16666666666666666\right) \cdot 2\right) \cdot J \]

   if -4.0000000000000001e30 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 1.99999999999999989e183

   1. Initial program 72.2%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)} \]
   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 l around 0

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

   7. Applied rewrites 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in K around 0

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

   12. Applied rewrites 86.6%

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

Alternative 13: 72.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

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 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 l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 88.5

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

7. Applied rewrites 88.5%

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

8. Taylor expanded in K around 0

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

10. Applied rewrites 72.5%

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

Alternative 14: 54.9% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

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 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 l around 0

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

7. Applied rewrites 65.3%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 65.3

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

9. Applied rewrites 65.3%

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

10. Taylor expanded in K around 0

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

12. Applied rewrites 54.9%

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

Alternative 15: 54.8% 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 85.7%

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

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 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

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
6. 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-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sinh.f64 81.0

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

7. Applied rewrites 81.0%

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

8. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f64 54.8

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

10. Applied rewrites 54.8%

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

Alternative 16: 46.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell + \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 80000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (+ l l) J)))
   (if (<= l -3.7e-38) t_0 (if (<= l 80000.0) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l + l) * J;
	double tmp;
	if (l <= -3.7e-38) {
		tmp = t_0;
	} else if (l <= 80000.0) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l + l) * j
    if (l <= (-3.7d-38)) then
        tmp = t_0
    else if (l <= 80000.0d0) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (l + l) * J;
	double tmp;
	if (l <= -3.7e-38) {
		tmp = t_0;
	} else if (l <= 80000.0) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (l + l) * J
	tmp = 0
	if l <= -3.7e-38:
		tmp = t_0
	elif l <= 80000.0:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l + l) * J)
	tmp = 0.0
	if (l <= -3.7e-38)
		tmp = t_0;
	elseif (l <= 80000.0)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l + l) * J;
	tmp = 0.0;
	if (l <= -3.7e-38)
		tmp = t_0;
	elseif (l <= 80000.0)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -3.7e-38], t$95$0, If[LessEqual[l, 80000.0], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.7e-38 or 8e4 < l

   1. Initial program 97.9%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in K around 0

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

      1. count-2-rev N/A

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

      2. lift-+.f64 22.1

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

   10. Applied rewrites 22.1%

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

   if -3.7e-38 < l < 8e4

   1. Initial program 72.7%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 71.2%

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

Alternative 17: 37.4% 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 85.7%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.4%

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

Reproduce

herbie shell --seed 2025111 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))