Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.9%

Time: 4.2s

Alternatives: 12

Speedup: 1.2×

• 50.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 86.6% accurate, 1.0× speedup

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, \sinh \ell \cdot J, U\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot \cos \left(-0.5 \cdot K\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, e^{\log \left(\frac{2}{\left(J + J\right) \cdot \sinh \ell}\right) \cdot -1}, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma 2.0 (* (sinh l) J) U)
     (if (<= t_0 4e+284)
       (fma (* (* l J) (cos (* -0.5 K))) 2.0 U)
       (fma 2.0 (exp (* (log (/ 2.0 (* (+ J J) (sinh l)))) -1.0)) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(2.0, (sinh(l) * J), U);
	} else if (t_0 <= 4e+284) {
		tmp = fma(((l * J) * cos((-0.5 * K))), 2.0, U);
	} else {
		tmp = fma(2.0, exp((log((2.0 / ((J + J) * sinh(l)))) * -1.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(2.0, Float64(sinh(l) * J), U);
	elseif (t_0 <= 4e+284)
		tmp = fma(Float64(Float64(l * J) * cos(Float64(-0.5 * K))), 2.0, U);
	else
		tmp = fma(2.0, exp(Float64(log(Float64(2.0 / Float64(Float64(J + J) * sinh(l)))) * -1.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 4e+284], N[(N[(N[(l * J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(2.0 * N[Exp[N[(N[Log[N[(2.0 / N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 4.0000000000000003e284

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 63.5%

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

   if 4.0000000000000003e284 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. sinh-def N/A

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

      4. associate-*l/ N/A

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

      5. sinh-undef N/A

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

      6. lift-sinh.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. div-flip N/A

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-/.f64 80.6%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.6%

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

   8. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-log.f64 46.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 46.0%

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

   10. Applied rewrites 46.0%

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{e^{\log \left(\frac{2}{\left(J + J\right) \cdot \sinh \ell}\right) \cdot -1}}, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, \sinh \ell \cdot J, U\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, e^{\log \left(\frac{2}{\left(J + J\right) \cdot \sinh \ell}\right) \cdot -1}, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma 2.0 (* (sinh l) J) U)
     (if (<= t_0 4e+284)
       (fma (* (+ J J) l) (cos (* -0.5 K)) U)
       (fma 2.0 (exp (* (log (/ 2.0 (* (+ J J) (sinh l)))) -1.0)) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(2.0, (sinh(l) * J), U);
	} else if (t_0 <= 4e+284) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(2.0, exp((log((2.0 / ((J + J) * sinh(l)))) * -1.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(2.0, Float64(sinh(l) * J), U);
	elseif (t_0 <= 4e+284)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(2.0, exp(Float64(log(Float64(2.0 / Float64(Float64(J + J) * sinh(l)))) * -1.0)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 4e+284], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(2.0 * N[Exp[N[(N[Log[N[(2.0 / N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 4.0000000000000003e284

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 63.4%

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

   if 4.0000000000000003e284 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. sinh-def N/A

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

      4. associate-*l/ N/A

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

      5. sinh-undef N/A

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

      6. lift-sinh.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. div-flip N/A

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-/.f64 80.6%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.6%

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

   8. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-log.f64 46.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 46.0%

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

   10. Applied rewrites 46.0%

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{e^{\log \left(\frac{2}{\left(J + J\right) \cdot \sinh \ell}\right) \cdot -1}}, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.46:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\sinh \ell \cdot 2\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.46)
   (fma (* (fma (* K K) -0.125 1.0) (* (sinh l) 2.0)) J 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)) <= -0.46) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (sinh(l) * 2.0)), J, 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)) <= -0.46)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(sinh(l) * 2.0)), J, 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], -0.46], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * J + 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))) < -0.46000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 68.8%

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

   if -0.46000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 80.7%

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

   8. Applied rewrites 80.7%

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

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.46:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.46)
   (fma (* (+ J J) (sinh 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)) <= -0.46) {
		tmp = fma(((J + J) * sinh(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)) <= -0.46)
		tmp = fma(Float64(Float64(J + J) * sinh(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], -0.46], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.46000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.4%

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

   6. Applied rewrites 69.2%

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

   if -0.46000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 80.7%

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

   8. Applied rewrites 80.7%

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

Alternative 6: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.46:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.46)
   (fma (* 2.0 (* J l)) (fma (* K K) -0.125 1.0) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.46) {
		tmp = fma((2.0 * (J * l)), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.46)
		tmp = fma(Float64(2.0 * Float64(J * l)), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.46], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.46000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.4%

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

   6. Applied rewrites 69.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 48.1%

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

   9. Applied rewrites 48.1%

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

   if -0.46000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 80.7%

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

   8. Applied rewrites 80.7%

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

Alternative 7: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.46:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sinh \ell \cdot J, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.46)
   (fma (* 2.0 (* J l)) (fma (* K K) -0.125 1.0) U)
   (fma 2.0 (* (sinh l) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.46) {
		tmp = fma((2.0 * (J * l)), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(2.0, (sinh(l) * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.46)
		tmp = fma(Float64(2.0 * Float64(J * l)), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(2.0, Float64(sinh(l) * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.46], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.46000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.4%

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

   6. Applied rewrites 69.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 48.1%

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

   9. Applied rewrites 48.1%

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

   if -0.46000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

Alternative 8: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.46:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(J \cdot \ell\right) \cdot 2}{U}\right) \cdot U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.46)
   (fma (* 2.0 (* J l)) (fma (* K K) -0.125 1.0) U)
   (* (+ 1.0 (/ (* (* J l) 2.0) U)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.46) {
		tmp = fma((2.0 * (J * l)), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = (1.0 + (((J * l) * 2.0) / U)) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.46)
		tmp = fma(Float64(2.0 * Float64(J * l)), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(J * l) * 2.0) / U)) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.46], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(J * l), $MachinePrecision] * 2.0), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.46000000000000002

   1. Initial program 86.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.4%

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

   6. Applied rewrites 69.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 48.1%

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

   9. Applied rewrites 48.1%

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

   if -0.46000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.7%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 53.8%

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

   9. Applied rewrites 53.8%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

       4. lower-unsound-*.f64 N/A

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

       5. lower-unsound-+.f64 N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 56.9%

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

   11. Applied rewrites 56.9%

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

Alternative 9: 56.9% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (* (+ 1.0 (/ (* (* J l) 2.0) U)) U))

C

double code(double J, double l, double K, double U) {
	return (1.0 + (((J * l) * 2.0) / U)) * 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 = (1.0d0 + (((j * l) * 2.0d0) / u)) * u
end function

Java

public static double code(double J, double l, double K, double U) {
	return (1.0 + (((J * l) * 2.0) / U)) * U;
}

Python

def code(J, l, K, U):
	return (1.0 + (((J * l) * 2.0) / U)) * U

Julia

function code(J, l, K, U)
	return Float64(Float64(1.0 + Float64(Float64(Float64(J * l) * 2.0) / U)) * U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = (1.0 + (((J * l) * 2.0) / U)) * U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(1.0 + N[(N[(N[(J * l), $MachinePrecision] * 2.0), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.7%

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

7. Taylor expanded in l around 0

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

   1. lower-*.f64 53.8%

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

9. Applied rewrites 53.8%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. sum-to-mult N/A

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

    4. lower-unsound-*.f64 N/A

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

    5. lower-unsound-+.f64 N/A

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

    6. lower-unsound-/.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 56.9%

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

11. Applied rewrites 56.9%

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

Alternative 10: 53.8% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (fma (* (* 2.0 1.0) J) l U))

C

double code(double J, double l, double K, double U) {
	return fma(((2.0 * 1.0) * J), l, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(2.0 * 1.0) * J), l, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(2.0 * 1.0), $MachinePrecision] * J), $MachinePrecision] * l + U), $MachinePrecision]

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in l around 0

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

6. Applied rewrites 63.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in K around 0

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

11. Applied rewrites 53.8%

    \[\leadsto \mathsf{fma}\left(\left(2 \cdot \color{blue}{1}\right) \cdot J, \ell, U\right) \]
12. Add Preprocessing

Alternative 11: 53.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.7%

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

7. Taylor expanded in l around 0

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

   1. lower-*.f64 53.8%

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

9. Applied rewrites 53.8%

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

Alternative 12: 37.2% accurate, 68.7× speedup

Math

\[U \]

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 86.6%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2025191 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))