Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.9%

Time: 4.7s

Alternatives: 17

Speedup: 1.2×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sinh \ell\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (+ J J) (cos (* 0.5 K))) (sinh l)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* J (- (exp l) (exp (- l)))) t_1)))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 2e+124)
       (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_1) U)
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J + J) * cos((0.5 * K))) * sinh(l);
	double t_1 = cos((K / 2.0));
	double t_2 = (J * (exp(l) - exp(-l))) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= 2e+124) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_1) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J + J) * cos(Float64(0.5 * K))) * sinh(l))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= 2e+124)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_1) + U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, 2e+124], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -inf.0 or 1.9999999999999999e124 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.7%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 99.8

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sinh.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sinh-undef-rev N/A

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

      8. rec-exp N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. rec-exp N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sinh-undef-rev N/A

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

   6. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 1.9999999999999999e124

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 3: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \left(t\_0 \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot J\\ t_2 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -390:\\ \;\;\;\;\left(1 - t\_2\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.068:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot J, 2 \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - t\_2\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (* (* t_0 (* (fma (* l l) 0.3333333333333333 2.0) l)) J))
        (t_2 (exp (- l))))
   (if (<= l -6.2e+91)
     t_1
     (if (<= l -390.0)
       (* (- 1.0 t_2) J)
       (if (<= l 0.068)
         (fma (* t_0 J) (* 2.0 l) U)
         (if (<= l 8.2e+102)
           (+ (* (* J (- (exp l) t_2)) (fma (* K K) -0.125 1.0)) U)
           t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = (t_0 * (fma((l * l), 0.3333333333333333, 2.0) * l)) * J;
	double t_2 = exp(-l);
	double tmp;
	if (l <= -6.2e+91) {
		tmp = t_1;
	} else if (l <= -390.0) {
		tmp = (1.0 - t_2) * J;
	} else if (l <= 0.068) {
		tmp = fma((t_0 * J), (2.0 * l), U);
	} else if (l <= 8.2e+102) {
		tmp = ((J * (exp(l) - t_2)) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(Float64(t_0 * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * J)
	t_2 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -6.2e+91)
		tmp = t_1;
	elseif (l <= -390.0)
		tmp = Float64(Float64(1.0 - t_2) * J);
	elseif (l <= 0.068)
		tmp = fma(Float64(t_0 * J), Float64(2.0 * l), U);
	elseif (l <= 8.2e+102)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - t_2)) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -6.2e+91], t$95$1, If[LessEqual[l, -390.0], N[(N[(1.0 - t$95$2), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 0.068], N[(N[(t$95$0 * J), $MachinePrecision] * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 8.2e+102], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.19999999999999995e91 or 8.1999999999999999e102 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 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 -6.19999999999999995e91 < l < -390

   1. Initial program 99.8%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 99.8

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

   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 76.9

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

   7. Applied rewrites 76.9%

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

      1. lift-sinh.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sinh-undef-rev N/A

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

      5. rec-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 76.9

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

   9. Applied rewrites 76.9%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 76.9%

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

   if -390 < l < 0.068000000000000005

   1. Initial program 72.2%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 99.0%

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

   if 0.068000000000000005 < l < 8.1999999999999999e102

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 75.5

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

   4. Applied rewrites 75.5%

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

Alternative 4: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.971:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.971)
     (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_0) U)
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.971) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.971)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.971], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$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))) < 0.970999999999999974

   1. Initial program 86.2%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.5

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

   4. Applied rewrites 87.5%

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

   if 0.970999999999999974 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 5: 88.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.8415:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), J \cdot \left(\ell + \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.8415)
     (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
     (if (<= t_0 -0.01)
       (fma (cos (* 0.5 K)) (* J (+ l l)) U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.8415) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= -0.01) {
		tmp = fma(cos((0.5 * K)), (J * (l + l)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.8415)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= -0.01)
		tmp = fma(cos(Float64(0.5 * K)), Float64(J * Float64(l + l)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.8415], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.841500000000000026

   1. Initial program 84.8%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.6

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

   4. Applied rewrites 63.6%

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

   if -0.841500000000000026 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 87.2%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 63.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 63.8

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 63.8

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

   9. Applied rewrites 63.8%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 6: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh \ell\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.25:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot t\_0\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh l))))
   (if (<= (cos (/ K 2.0)) -0.25)
     (* (* (* (* K K) -0.125) 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 (cos((K / 2.0)) <= -0.25) {
		tmp = (((K * K) * -0.125) * 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 (cos(Float64(K / 2.0)) <= -0.25)
		tmp = Float64(Float64(Float64(Float64(K * K) * -0.125) * 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[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.25], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] * t$95$0), $MachinePrecision] * J), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.4%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 50.2

         \[\leadsto \left(\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

   10. Applied rewrites 50.2%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot -0.125\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J \]

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

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 92.8

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

   4. Applied rewrites 92.8%

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

Alternative 8: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), J \cdot \left(\ell + \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.75)
   (fma (fma -0.125 (* K K) 1.0) (* J (+ l l)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.75) {
		tmp = fma(fma(-0.125, (K * K), 1.0), (J * (l + l)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.75)
		tmp = fma(fma(-0.125, Float64(K * K), 1.0), Float64(J * Float64(l + l)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.75], N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(J * N[(l + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 85.7%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 63.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 63.4

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 63.4

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

   9. Applied rewrites 63.4%

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

   10. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 49.2

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

   12. Applied rewrites 49.2%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 86.6

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

   4. Applied rewrites 86.6%

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

Alternative 9: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 - t\_0\right) \cdot J\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - 1\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (- 1.0 t_0) J)
     (if (<= t_1 20.0)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)
       (* (- (exp l) 1.0) J)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 - t_0) * J;
	} else if (t_1 <= 20.0) {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	} else {
		tmp = (exp(l) - 1.0) * J;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - t_0) * J);
	elseif (t_1 <= 20.0)
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	else
		tmp = Float64(Float64(exp(l) - 1.0) * J);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 - t$95$0), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Exp[l], $MachinePrecision] - 1.0), $MachinePrecision] * J), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in 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 74.4

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

   7. Applied rewrites 74.4%

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

      1. lift-sinh.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sinh-undef-rev N/A

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

      5. rec-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 74.4

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

   9. Applied rewrites 74.4%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 74.4%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 20

   1. Initial program 72.3%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 85.7

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

   10. Applied rewrites 85.7%

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

   if 20 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 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 75.9

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

   7. Applied rewrites 75.9%

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

      1. lift-sinh.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sinh-undef-rev N/A

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

      5. rec-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 75.9

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 75.8%

       \[\leadsto \left(e^{\ell} - 1\right) \cdot J \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;\mathsf{fma}\left(K \cdot \left(K \cdot \ell\right), -0.25, \ell + \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.75)
   (* (fma (* K (* K l)) -0.25 (+ l l)) J)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.75) {
		tmp = fma((K * (K * l)), -0.25, (l + l)) * J;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.75)
		tmp = Float64(fma(Float64(K * Float64(K * l)), -0.25, Float64(l + l)) * J);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.75], N[(N[(N[(K * N[(K * l), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(l + l), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 85.7%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 64.6

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 28.2

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 41.6

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

   10. Applied rewrites 41.6%

       \[\leadsto \mathsf{fma}\left(\left(K \cdot K\right) \cdot \ell, -0.25, \ell + \ell\right) \cdot J \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 41.7

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

   12. Applied rewrites 41.7%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 11: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.75)
   (* (* (* (* K K) l) -0.25) J)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.75) {
		tmp = (((K * K) * l) * -0.25) * J;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.75)
		tmp = Float64(Float64(Float64(Float64(K * K) * l) * -0.25) * J);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.75], N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * -0.25), $MachinePrecision] * J), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 85.7%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 64.6

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 28.2

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 41.6

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

   10. Applied rewrites 41.6%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 41.7

          \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J \]

   13. Applied rewrites 41.7%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J \]

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 12: 72.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{elif}\;\ell \leq 15.5:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - 1\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5e+31)
   (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
   (if (<= l 15.5) (fma (+ l l) J U) (* (- (exp l) 1.0) J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5e+31) {
		tmp = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	} else if (l <= 15.5) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = (exp(l) - 1.0) * J;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5e+31)
		tmp = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J);
	elseif (l <= 15.5)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = Float64(Float64(exp(l) - 1.0) * J);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5e+31], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 15.5], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Exp[l], $MachinePrecision] - 1.0), $MachinePrecision] * J), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.00000000000000027e31

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in 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%

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

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.1

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

   10. Applied rewrites 61.1%

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

   if -5.00000000000000027e31 < l < 15.5

   1. Initial program 73.5%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 85.7

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 82.0

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

   10. Applied rewrites 82.0%

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

   if 15.5 < l

   1. Initial program 99.9%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 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 75.9

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

   7. Applied rewrites 75.9%

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

      1. lift-sinh.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sinh-undef-rev N/A

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

      5. rec-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 75.9

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 75.9%

       \[\leadsto \left(e^{\ell} - 1\right) \cdot J \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)))
   (if (<= l -5e+31) t_0 (if (<= l 1e+32) (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 <= -5e+31) {
		tmp = t_0;
	} else if (l <= 1e+32) {
		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 <= -5e+31)
		tmp = t_0;
	elseif (l <= 1e+32)
		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, -5e+31], t$95$0, If[LessEqual[l, 1e+32], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000000027e31 or 1.00000000000000005e32 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in 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 74.9

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

   7. Applied rewrites 74.9%

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

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.0

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

   10. Applied rewrites 62.0%

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

   if -5.00000000000000027e31 < l < 1.00000000000000005e32

   1. Initial program 74.6%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 85.3

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 78.5

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

   10. Applied rewrites 78.5%

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

Alternative 14: 55.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.75)
   (* (* (* (* K K) l) -0.25) J)
   (fma (+ l l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.75) {
		tmp = (((K * K) * l) * -0.25) * J;
	} else {
		tmp = fma((l + l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.75)
		tmp = Float64(Float64(Float64(Float64(K * K) * l) * -0.25) * J);
	else
		tmp = fma(Float64(l + l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.75], N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * -0.25), $MachinePrecision] * J), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 85.7%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 64.6

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 28.2

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 41.6

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

   10. Applied rewrites 41.6%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 41.7

          \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J \]

   13. Applied rewrites 41.7%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot -0.25\right) \cdot J \]

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   2. Taylor expanded in J around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 56.9

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

   10. Applied rewrites 56.9%

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

Alternative 15: 54.5% 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 86.0%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sinh-undef-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sinh.f64 80.6

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

7. Applied rewrites 80.6%

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

8. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f64 54.5

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

10. Applied rewrites 54.5%

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

Alternative 16: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \left(\ell + \ell\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+124}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* (+ l l) J)))
   (if (<= t_0 -4e+231) t_1 (if (<= t_0 2e+124) 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 = (l + l) * J;
	double tmp;
	if (t_0 <= -4e+231) {
		tmp = t_1;
	} else if (t_0 <= 2e+124) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    t_1 = (l + l) * j
    if (t_0 <= (-4d+231)) then
        tmp = t_1
    else if (t_0 <= 2d+124) then
        tmp = u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double t_1 = (l + l) * J;
	double tmp;
	if (t_0 <= -4e+231) {
		tmp = t_1;
	} else if (t_0 <= 2e+124) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	t_1 = (l + l) * J
	tmp = 0
	if t_0 <= -4e+231:
		tmp = t_1
	elif t_0 <= 2e+124:
		tmp = 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(l + l) * J)
	tmp = 0.0
	if (t_0 <= -4e+231)
		tmp = t_1;
	elseif (t_0 <= 2e+124)
		tmp = U;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	t_1 = (l + l) * J;
	tmp = 0.0;
	if (t_0 <= -4e+231)
		tmp = t_1;
	elseif (t_0 <= 2e+124)
		tmp = U;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(l + l), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+231], t$95$1, If[LessEqual[t$95$0, 2e+124], U, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.0000000000000002e231 or 1.9999999999999999e124 < (*.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. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sinh-undef N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sinh.f64 99.8

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

   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 l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

   8. Taylor expanded in K around 0

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

      1. count-2-rev N/A

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

      2. lift-+.f64 23.0

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

   10. Applied rewrites 23.0%

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

   if -4.0000000000000002e231 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1.9999999999999999e124

   1. Initial program 72.3%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 70.5%

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

Alternative 17: 36.3% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.3%

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

Reproduce

herbie shell --seed 2025116 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))