Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.4% → 99.9%

Time: 4.8s

Alternatives: 17

Speedup: 1.3×

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \left(\left(J + J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+220}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l)))))
        (t_1 (* (* (+ J J) (cos (* K 0.5))) (sinh l))))
   (if (<= t_0 -5e-70)
     t_1
     (if (<= t_0 1e+220)
       (+
        (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0)))
        U)
       t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = ((J + J) * cos((K * 0.5))) * sinh(l);
	double tmp;
	if (t_0 <= -5e-70) {
		tmp = t_1;
	} else if (t_0 <= 1e+220) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(Float64(Float64(J + J) * cos(Float64(K * 0.5))) * sinh(l))
	tmp = 0.0
	if (t_0 <= -5e-70)
		tmp = t_1;
	elseif (t_0 <= 1e+220)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-70], t$95$1, If[LessEqual[t$95$0, 1e+220], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.9999999999999998e-70 or 1e220 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.6%

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

   2. Taylor expanded in J around 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.4

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

   4. Applied rewrites 99.4%

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

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

      16. associate-*r* N/A

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

   6. Applied rewrites 99.4%

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

   if -4.9999999999999998e-70 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1e220

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 2: 99.6% 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 85.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot J\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -82:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.27:\\ \;\;\;\;\left(J \cdot \left(\ell + \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (* (* (cos (* 0.5 K)) (* (fma (* l l) 0.3333333333333333 2.0) l)) J)))
   (if (<= l -9.2e+91)
     t_0
     (if (<= l -82.0)
       (* (* (sinh l) 2.0) J)
       (if (<= l 0.27)
         (+ (* (* J (+ l l)) (cos (/ K 2.0))) U)
         (if (<= l 1.5e+98)
           (* (* (fma -0.125 (* K K) 1.0) (* 2.0 (sinh l))) J)
           t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (cos((0.5 * K)) * (fma((l * l), 0.3333333333333333, 2.0) * l)) * J;
	double tmp;
	if (l <= -9.2e+91) {
		tmp = t_0;
	} else if (l <= -82.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 0.27) {
		tmp = ((J * (l + l)) * cos((K / 2.0))) + U;
	} else if (l <= 1.5e+98) {
		tmp = (fma(-0.125, (K * K), 1.0) * (2.0 * sinh(l))) * J;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(cos(Float64(0.5 * K)) * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * J)
	tmp = 0.0
	if (l <= -9.2e+91)
		tmp = t_0;
	elseif (l <= -82.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 0.27)
		tmp = Float64(Float64(Float64(J * Float64(l + l)) * cos(Float64(K / 2.0))) + U);
	elseif (l <= 1.5e+98)
		tmp = Float64(Float64(fma(-0.125, Float64(K * K), 1.0) * Float64(2.0 * sinh(l))) * J);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -9.2e+91], t$95$0, If[LessEqual[l, -82.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.27], N[(N[(N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.5e+98], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.19999999999999965e91 or 1.5000000000000001e98 < 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. 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 98.5

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

   7. Applied rewrites 98.5%

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

   if -9.19999999999999965e91 < l < -82

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.1

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

   4. Applied rewrites 99.1%

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

   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 74.2

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

   7. Applied rewrites 74.2%

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

   if -82 < l < 0.27000000000000002

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 0.27000000000000002 < l < 1.5000000000000001e98

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      3. mult-flip N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      4. +-commutative N/A

         \[\leadsto \left(\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      7. sub-flip N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \frac{-1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      13. lift-*.f64 68.1

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   7. Applied rewrites 68.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 72.6%

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

Alternative 4: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 300000:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\sinh \ell \cdot 2\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 300000.0)
   (fma J (* (* (sinh l) 2.0) (fma (* K K) -0.125 1.0)) U)
   (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 300000.0) {
		tmp = fma(J, ((sinh(l) * 2.0) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 300000.0)
		tmp = fma(J, Float64(Float64(sinh(l) * 2.0) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 300000.0], N[(J * N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 3e5

   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 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 78.7%

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

   if 3e5 < K

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

Alternative 5: 87.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -82:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.27:\\ \;\;\;\;\left(J \cdot \left(\ell + \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot t\_0\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= l -82.0)
     (* (* (sinh l) 2.0) J)
     (if (<= l 0.27)
       (+ (* (* J (+ l l)) (cos (/ K 2.0))) U)
       (if (<= l 4.5e+33)
         (* (* (fma -0.125 (* K K) 1.0) t_0) J)
         (fma t_0 J U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (l <= -82.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 0.27) {
		tmp = ((J * (l + l)) * cos((K / 2.0))) + U;
	} else if (l <= 4.5e+33) {
		tmp = (fma(-0.125, (K * K), 1.0) * t_0) * J;
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (l <= -82.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 0.27)
		tmp = Float64(Float64(Float64(J * Float64(l + l)) * cos(Float64(K / 2.0))) + U);
	elseif (l <= 4.5e+33)
		tmp = Float64(Float64(fma(-0.125, Float64(K * K), 1.0) * t_0) * J);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -82.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.27], N[(N[(N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 4.5e+33], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * J), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -82

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   if -82 < l < 0.27000000000000002

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 0.27000000000000002 < l < 4.5e33

   1. Initial program 99.6%

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

   2. Taylor expanded in J around 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      3. mult-flip N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      4. +-commutative N/A

         \[\leadsto \left(\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      7. sub-flip N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \frac{-1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      13. lift-*.f64 66.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   7. Applied rewrites 66.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 67.2%

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

   if 4.5e33 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 75.1

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

   4. Applied rewrites 75.1%

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

Alternative 6: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -82:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot t\_0\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= l -82.0)
     (* (* (sinh l) 2.0) J)
     (if (<= l 0.27)
       (fma (* (+ l l) J) (cos (* K 0.5)) U)
       (if (<= l 4.5e+33)
         (* (* (fma -0.125 (* K K) 1.0) t_0) J)
         (fma t_0 J U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (l <= -82.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 0.27) {
		tmp = fma(((l + l) * J), cos((K * 0.5)), U);
	} else if (l <= 4.5e+33) {
		tmp = (fma(-0.125, (K * K), 1.0) * t_0) * J;
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (l <= -82.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 0.27)
		tmp = fma(Float64(Float64(l + l) * J), cos(Float64(K * 0.5)), U);
	elseif (l <= 4.5e+33)
		tmp = Float64(Float64(fma(-0.125, Float64(K * K), 1.0) * t_0) * J);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -82.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.27], N[(N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 4.5e+33], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * J), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -82

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   if -82 < l < 0.27000000000000002

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      14. sinh-undef-rev N/A

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

      15. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(K \cdot \frac{1}{2}\right)\right), U\right) \]

   6. Applied rewrites 99.3%

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

   if 0.27000000000000002 < l < 4.5e33

   1. Initial program 99.6%

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

   2. Taylor expanded in J around 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      3. mult-flip N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      4. +-commutative N/A

         \[\leadsto \left(\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      7. sub-flip N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \frac{-1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      13. lift-*.f64 66.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   7. Applied rewrites 66.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 67.2%

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

   if 4.5e33 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 75.1

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

   4. Applied rewrites 75.1%

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

Alternative 7: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -82:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, \cos \left(0.5 \cdot K\right) \cdot J, U\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot t\_0\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= l -82.0)
     (* (* (sinh l) 2.0) J)
     (if (<= l 0.27)
       (fma (+ l l) (* (cos (* 0.5 K)) J) U)
       (if (<= l 4.5e+33)
         (* (* (fma -0.125 (* K K) 1.0) t_0) J)
         (fma t_0 J U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (l <= -82.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 0.27) {
		tmp = fma((l + l), (cos((0.5 * K)) * J), U);
	} else if (l <= 4.5e+33) {
		tmp = (fma(-0.125, (K * K), 1.0) * t_0) * J;
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (l <= -82.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 0.27)
		tmp = fma(Float64(l + l), Float64(cos(Float64(0.5 * K)) * J), U);
	elseif (l <= 4.5e+33)
		tmp = Float64(Float64(fma(-0.125, Float64(K * K), 1.0) * t_0) * J);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -82.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.27], N[(N[(l + l), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 4.5e+33], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * J), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -82

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   if -82 < l < 0.27000000000000002

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      14. sinh-undef-rev N/A

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

      15. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(K \cdot \frac{1}{2}\right)\right), U\right) \]

   6. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite=>}\left(lower-*.f64, \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)\right), U\right) \]

      11. *-commutative N/A

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

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot J, U\right) \]

      13. *-commutative N/A

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

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot K\right)\right) \cdot J, U\right) \]

   8. Applied rewrites 99.3%

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

   if 0.27000000000000002 < l < 4.5e33

   1. Initial program 99.6%

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

   2. Taylor expanded in J around 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      3. mult-flip N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      4. +-commutative N/A

         \[\leadsto \left(\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      7. sub-flip N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \frac{-1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      13. lift-*.f64 66.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   7. Applied rewrites 66.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 67.2%

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

   if 4.5e33 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 75.1

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

   4. Applied rewrites 75.1%

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

Alternative 8: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\ell \leq -82:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot t\_0\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= l -82.0)
     (* (* (sinh l) 2.0) J)
     (if (<= l 0.27)
       (fma (+ J J) (* (cos (* 0.5 K)) l) U)
       (if (<= l 4.5e+33)
         (* (* (fma -0.125 (* K K) 1.0) t_0) J)
         (fma t_0 J U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * sinh(l);
	double tmp;
	if (l <= -82.0) {
		tmp = (sinh(l) * 2.0) * J;
	} else if (l <= 0.27) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), U);
	} else if (l <= 4.5e+33) {
		tmp = (fma(-0.125, (K * K), 1.0) * t_0) * J;
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (l <= -82.0)
		tmp = Float64(Float64(sinh(l) * 2.0) * J);
	elseif (l <= 0.27)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), U);
	elseif (l <= 4.5e+33)
		tmp = Float64(Float64(fma(-0.125, Float64(K * K), 1.0) * t_0) * J);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -82.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.27], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 4.5e+33], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * J), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -82

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   if -82 < l < 0.27000000000000002

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-cos.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 0.27000000000000002 < l < 4.5e33

   1. Initial program 99.6%

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

   2. Taylor expanded in J around 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      3. mult-flip N/A

         \[\leadsto \left(\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      4. +-commutative N/A

         \[\leadsto \left(\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2} + 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      7. sub-flip N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{384} \cdot {K}^{2} + \frac{-1}{8}, {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), {K}^{2}, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{384}, K \cdot K, \frac{-1}{8}\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

      13. lift-*.f64 66.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   7. Applied rewrites 66.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), K \cdot K, 1\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 67.2%

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

   if 4.5e33 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 75.1

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

   4. Applied rewrites 75.1%

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

Alternative 9: 87.0% 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 65.4

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

   4. Applied rewrites 65.4%

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

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

   1. Initial program 85.5%

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

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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: 86.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma J (* (* (sinh l) 2.0) (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 = fma(J, ((sinh(l) * 2.0) * 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 = fma(J, Float64(Float64(sinh(l) * 2.0) * 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[(J * N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $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 65.4

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

   4. Applied rewrites 65.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 60.8%

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

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

   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

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 11: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+
    (* (* (* (* (* l l) l) J) 0.3333333333333333) (fma (* K K) -0.125 1.0))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = (((((l * l) * l) * J) * 0.3333333333333333) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.3333333333333333) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 61.6

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

   10. Applied rewrites 61.6%

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

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

   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

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 12: 81.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* (+ 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)) <= -0.01) {
		tmp = fma(((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)) <= -0.01)
		tmp = fma(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], -0.01], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -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 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in l around 0

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

      1. sinh-undef-rev N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 50.7

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

   10. Applied rewrites 50.7%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lower-fma.f64 50.7

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

   12. Applied rewrites 50.7%

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

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

   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

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   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 13: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \left(\sinh \ell \cdot 2\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* (* (sinh l) 2.0) J)))
   (if (<= t_0 (- INFINITY)) t_1 (if (<= t_0 0.0002) (fma (+ l l) J U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = (sinh(l) * 2.0) * J;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.0002) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(Float64(sinh(l) * 2.0) * J)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.0002)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.0002], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 74.0

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

   7. Applied rewrites 74.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-4

   1. Initial program 71.5%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      14. sinh-undef-rev N/A

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

      15. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(K \cdot \frac{1}{2}\right)\right), U\right) \]

   6. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite=>}\left(lower-*.f64, \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)\right), U\right) \]

      11. *-commutative N/A

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

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot J, U\right) \]

      13. *-commutative N/A

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

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot K\right)\right) \cdot J, U\right) \]

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 85.5%

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

Alternative 14: 72.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -9200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.37:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)))
   (if (<= l -9200.0) t_0 (if (<= l 0.37) (fma (+ l l) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	double tmp;
	if (l <= -9200.0) {
		tmp = t_0;
	} else if (l <= 0.37) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J)
	tmp = 0.0
	if (l <= -9200.0)
		tmp = t_0;
	elseif (l <= 0.37)
		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[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -9200.0], t$95$0, If[LessEqual[l, 0.37], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -9200 or 0.37 < 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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in 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 74.2

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if -9200 < l < 0.37

   1. Initial program 71.6%

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

   2. Taylor expanded in l around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(\frac{1}{2} \cdot K\right)\right), U\right) \]

      14. sinh-undef-rev N/A

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

      15. sinh-undef-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(K \cdot \frac{1}{2}\right)\right), U\right) \]

   6. Applied rewrites 99.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite=>}\left(lower-*.f64, \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)\right), U\right) \]

      11. *-commutative N/A

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

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot J, U\right) \]

      13. *-commutative N/A

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

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot K\right)\right) \cdot J, U\right) \]

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 85.3%

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

Alternative 15: 54.2% 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 85.4%

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

2. Taylor expanded in l around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 64.8

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

4. Applied rewrites 64.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-undef-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(\frac{1}{2} \cdot K\right)\right), U\right) \]

   14. sinh-undef-rev N/A

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

   15. sinh-undef-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\ell} + \ell\right) \cdot J, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(K \cdot \frac{1}{2}\right)\right), U\right) \]

6. Applied rewrites 64.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite=>}\left(lower-*.f64, \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)\right), U\right) \]

   11. *-commutative N/A

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

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \mathsf{Rewrite<=}\left(lift-cos.f64, \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot J, U\right) \]

   13. *-commutative N/A

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

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\ell} + \ell, \cos \mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{1}{2} \cdot K\right)\right) \cdot J, U\right) \]

8. Applied rewrites 64.8%

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

9. Taylor expanded in K around 0

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

11. Applied rewrites 54.2%

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

Alternative 16: 45.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot 2\right) \cdot J\\ \mathbf{if}\;\ell \leq -1050:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* l 2.0) J)))
   (if (<= l -1050.0) t_0 (if (<= l 3.2e-25) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l * 2.0) * J;
	double tmp;
	if (l <= -1050.0) {
		tmp = t_0;
	} else if (l <= 3.2e-25) {
		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 * 2.0d0) * j
    if (l <= (-1050.0d0)) then
        tmp = t_0
    else if (l <= 3.2d-25) 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 * 2.0) * J;
	double tmp;
	if (l <= -1050.0) {
		tmp = t_0;
	} else if (l <= 3.2e-25) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (l * 2.0) * J
	tmp = 0
	if l <= -1050.0:
		tmp = t_0
	elif l <= 3.2e-25:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l * 2.0) * J)
	tmp = 0.0
	if (l <= -1050.0)
		tmp = t_0;
	elseif (l <= 3.2e-25)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l * 2.0) * J;
	tmp = 0.0;
	if (l <= -1050.0)
		tmp = t_0;
	elseif (l <= 3.2e-25)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * 2.0), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1050.0], t$95$0, If[LessEqual[l, 3.2e-25], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1050 or 3.2000000000000001e-25 < l

   1. Initial program 99.0%

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

   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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-cos.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 97.8

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

   4. Applied rewrites 97.8%

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

   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 72.5

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

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 21.1%

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

   if -1050 < l < 3.2000000000000001e-25

   1. Initial program 71.6%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 70.4%

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

Alternative 17: 37.0% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

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

4. Applied rewrites 37.0%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))