Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.9%

Time: 6.0s

Alternatives: 18

Speedup: 1.2×

• 50.1% 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: 86.7% 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 86.7%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

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: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(1 - e^{-\ell}\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 22:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \ell \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.0)
   (fma (* (cos (* -0.5 K)) (- 1.0 (exp (- l)))) J U)
   (if (<= l 22.0)
     (fma (cos (* 0.5 K)) (* l (+ J J)) U)
     (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.0) {
		tmp = fma((cos((-0.5 * K)) * (1.0 - exp(-l))), J, U);
	} else if (l <= 22.0) {
		tmp = fma(cos((0.5 * K)), (l * (J + J)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.0)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(1.0 - exp(Float64(-l)))), J, U);
	elseif (l <= 22.0)
		tmp = fma(cos(Float64(0.5 * K)), Float64(l * Float64(J + J)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.0], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 22.0], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if -4 < l < 22

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 63.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   10. Applied rewrites 63.3%

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

   if 22 < l

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 3: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := t\_0 \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+182}:\\ \;\;\;\;U + t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \ell \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (* t_0 (cos (/ K 2.0)))))
   (if (<= t_1 -1e+182)
     (+ U t_0)
     (if (<= t_1 2e+112)
       (fma (cos (* 0.5 K)) (* l (+ J J)) U)
       (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = t_0 * cos((K / 2.0));
	double tmp;
	if (t_1 <= -1e+182) {
		tmp = U + t_0;
	} else if (t_1 <= 2e+112) {
		tmp = fma(cos((0.5 * K)), (l * (J + J)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(t_0 * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= -1e+182)
		tmp = Float64(U + t_0);
	elseif (t_1 <= 2e+112)
		tmp = fma(cos(Float64(0.5 * K)), Float64(l * Float64(J + J)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+182], N[(U + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e+112], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -1.0000000000000001e182

   1. Initial program 86.7%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.2%

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

   4. Applied rewrites 73.2%

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

   if -1.0000000000000001e182 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 1.9999999999999999e112

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 63.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   10. Applied rewrites 63.3%

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

   if 1.9999999999999999e112 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 4: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(e^{\ell} \cdot J - e^{-\ell} \cdot J, t\_0, U\right)\\ \mathbf{elif}\;t\_1 \leq 0.9998:\\ \;\;\;\;U + \mathsf{fma}\left(-1, \frac{J}{e^{\ell}}, J \cdot e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, t\_0, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* K K) -0.125 1.0)) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.2)
     (fma (- (* (exp l) J) (* (exp (- l)) J)) t_0 U)
     (if (<= t_1 0.9998)
       (+ U (fma -1.0 (/ J (exp l)) (* J (exp l))))
       (fma (* (+ J J) (sinh l)) t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((K * K), -0.125, 1.0);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = fma(((exp(l) * J) - (exp(-l) * J)), t_0, U);
	} else if (t_1 <= 0.9998) {
		tmp = U + fma(-1.0, (J / exp(l)), (J * exp(l)));
	} else {
		tmp = fma(((J + J) * sinh(l)), t_0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(K * K), -0.125, 1.0)
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = fma(Float64(Float64(exp(l) * J) - Float64(exp(Float64(-l)) * J)), t_0, U);
	elseif (t_1 <= 0.9998)
		tmp = Float64(U + fma(-1.0, Float64(J / exp(l)), Float64(J * exp(l))));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), t_0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[(N[Exp[l], $MachinePrecision] * J), $MachinePrecision] - N[(N[Exp[(-l)], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * t$95$0 + U), $MachinePrecision], If[LessEqual[t$95$1, 0.9998], N[(U + N[(-1.0 * N[(J / N[Exp[l], $MachinePrecision]), $MachinePrecision] + N[(J * N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * t$95$0 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-rgt-in N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 N/A

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

   8. Applied rewrites 64.8%

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

   if -0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99980000000000002

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. exp-neg N/A

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

      17. distribute-neg-frac2 N/A

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

      18. mult-flip-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-exp.f64 64.8%

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

   8. Applied rewrites 64.8%

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

   9. Taylor expanded in K around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-exp.f64 73.2%

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

   11. Applied rewrites 73.2%

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

   if 0.99980000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 5: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(J, e^{\ell}, \frac{J}{-e^{\ell}}\right), t\_0, U\right)\\ \mathbf{elif}\;t\_1 \leq 0.9998:\\ \;\;\;\;U + \mathsf{fma}\left(-1, \frac{J}{e^{\ell}}, J \cdot e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, t\_0, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* K K) -0.125 1.0)) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.2)
     (fma (fma J (exp l) (/ J (- (exp l)))) t_0 U)
     (if (<= t_1 0.9998)
       (+ U (fma -1.0 (/ J (exp l)) (* J (exp l))))
       (fma (* (+ J J) (sinh l)) t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((K * K), -0.125, 1.0);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = fma(fma(J, exp(l), (J / -exp(l))), t_0, U);
	} else if (t_1 <= 0.9998) {
		tmp = U + fma(-1.0, (J / exp(l)), (J * exp(l)));
	} else {
		tmp = fma(((J + J) * sinh(l)), t_0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(K * K), -0.125, 1.0)
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = fma(fma(J, exp(l), Float64(J / Float64(-exp(l)))), t_0, U);
	elseif (t_1 <= 0.9998)
		tmp = Float64(U + fma(-1.0, Float64(J / exp(l)), Float64(J * exp(l))));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), t_0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(J * N[Exp[l], $MachinePrecision] + N[(J / (-N[Exp[l], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * t$95$0 + U), $MachinePrecision], If[LessEqual[t$95$1, 0.9998], N[(U + N[(-1.0 * N[(J / N[Exp[l], $MachinePrecision]), $MachinePrecision] + N[(J * N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * t$95$0 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. exp-neg N/A

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

      17. distribute-neg-frac2 N/A

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

      18. mult-flip-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-exp.f64 64.8%

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

   8. Applied rewrites 64.8%

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

   if -0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99980000000000002

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. exp-neg N/A

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

      17. distribute-neg-frac2 N/A

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

      18. mult-flip-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-exp.f64 64.8%

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

   8. Applied rewrites 64.8%

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

   9. Taylor expanded in K around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-exp.f64 73.2%

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

   11. Applied rewrites 73.2%

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

   if 0.99980000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 6: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{J \cdot \left(\ell \cdot \left(2 + 2 \cdot \ell\right)\right)}{e^{\ell}} + U\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;U + \mathsf{fma}\left(-1, \frac{J}{e^{\ell}}, J \cdot e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ (/ (* J (* l (+ 2.0 (* 2.0 l)))) (exp l)) U)
     (if (<= t_0 0.9998)
       (+ U (fma -1.0 (/ J (exp l)) (* J (exp l))))
       (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / exp(l)) + U;
	} else if (t_0 <= 0.9998) {
		tmp = U + fma(-1.0, (J / exp(l)), (J * exp(l)));
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(2.0 * l)))) / exp(l)) + U);
	elseif (t_0 <= 0.9998)
		tmp = Float64(U + fma(-1.0, Float64(J / exp(l)), Float64(J * exp(l))));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[(J * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.9998], N[(U + N[(-1.0 * N[(J / N[Exp[l], $MachinePrecision]), $MachinePrecision] + N[(J * N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.4%

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

   9. Applied rewrites 50.4%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99980000000000002

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. exp-neg N/A

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

      17. distribute-neg-frac2 N/A

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

      18. mult-flip-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-exp.f64 64.8%

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

   8. Applied rewrites 64.8%

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

   9. Taylor expanded in K around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-exp.f64 73.2%

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

   11. Applied rewrites 73.2%

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

   if 0.99980000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 7: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{J \cdot \left(\ell \cdot \left(2 + 2 \cdot \ell\right)\right)}{e^{\ell}} + U\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ (/ (* J (* l (+ 2.0 (* 2.0 l)))) (exp l)) U)
     (if (<= t_0 0.9998)
       (+ U (* J (- (exp l) (/ 1.0 (exp l)))))
       (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / exp(l)) + U;
	} else if (t_0 <= 0.9998) {
		tmp = U + (J * (exp(l) - (1.0 / exp(l))));
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(2.0 * l)))) / exp(l)) + U);
	elseif (t_0 <= 0.9998)
		tmp = Float64(U + Float64(J * Float64(exp(l) - Float64(1.0 / exp(l)))));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[(J * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.9998], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.4%

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

   9. Applied rewrites 50.4%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99980000000000002

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 73.2%

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

   6. Applied rewrites 73.2%

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

   if 0.99980000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 8: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{J \cdot \left(\ell \cdot \left(2 + 2 \cdot \ell\right)\right)}{e^{\ell}} + U\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ (/ (* J (* l (+ 2.0 (* 2.0 l)))) (exp l)) U)
     (if (<= t_0 0.9998)
       (+ U (* J (- (exp l) (exp (- l)))))
       (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / exp(l)) + U;
	} else if (t_0 <= 0.9998) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(2.0 * l)))) / exp(l)) + U);
	elseif (t_0 <= 0.9998)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[(J * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.9998], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.4%

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

   9. Applied rewrites 50.4%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99980000000000002

   1. Initial program 86.7%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.2%

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

   4. Applied rewrites 73.2%

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

   if 0.99980000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

Alternative 9: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\frac{J \cdot \left(\ell \cdot \left(2 + 2 \cdot \ell\right)\right)}{e^{\ell}} + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ (/ (* J (* l (+ 2.0 (* 2.0 l)))) (exp l)) U)
   (+ U (* J (- (exp l) (exp (- l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / exp(l)) + U;
	} else {
		tmp = U + (J * (exp(l) - exp(-l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = ((j * (l * (2.0d0 + (2.0d0 * l)))) / exp(l)) + u
    else
        tmp = u + (j * (exp(l) - exp(-l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / Math.exp(l)) + U;
	} else {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / math.exp(l)) + U
	else:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(2.0 * l)))) / exp(l)) + U);
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = ((J * (l * (2.0 + (2.0 * l)))) / exp(l)) + U;
	else
		tmp = U + (J * (exp(l) - exp(-l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(J * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.4%

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

   9. Applied rewrites 50.4%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.2%

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

   4. Applied rewrites 73.2%

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

Alternative 10: 80.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq 500:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} + U\\ \mathbf{else}:\\ \;\;\;\;\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \ell} + U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 500.0)
   (+ (/ (* (expm1 (+ l l)) J) (exp l)) U)
   (+ (/ (* J (expm1 (* 2.0 l))) (+ 1.0 l)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 500.0) {
		tmp = ((expm1((l + l)) * J) / exp(l)) + U;
	} else {
		tmp = ((J * expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 500.0) {
		tmp = ((Math.expm1((l + l)) * J) / Math.exp(l)) + U;
	} else {
		tmp = ((J * Math.expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 500.0:
		tmp = ((math.expm1((l + l)) * J) / math.exp(l)) + U
	else:
		tmp = ((J * math.expm1((2.0 * l))) / (1.0 + l)) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 500.0)
		tmp = Float64(Float64(Float64(expm1(Float64(l + l)) * J) / exp(l)) + U);
	else
		tmp = Float64(Float64(Float64(J * expm1(Float64(2.0 * l))) / Float64(1.0 + l)) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 500.0], N[(N[(N[(N[(Exp[N[(l + l), $MachinePrecision]] - 1), $MachinePrecision] * J), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(Exp[N[(2.0 * l), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 500

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]
   7. Step-by-step derivation

   8. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} + U} \]

   if 500 < l

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
   8. Step-by-step derivation

      1. lower-+.f64 62.4%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \color{blue}{\ell}} + U \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq 500:\\ \;\;\;\;\mathsf{fma}\left(e^{-\ell}, \mathsf{expm1}\left(\ell + \ell\right) \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \ell} + U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 500.0)
   (fma (exp (- l)) (* (expm1 (+ l l)) J) U)
   (+ (/ (* J (expm1 (* 2.0 l))) (+ 1.0 l)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 500.0) {
		tmp = fma(exp(-l), (expm1((l + l)) * J), U);
	} else {
		tmp = ((J * expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 500.0)
		tmp = fma(exp(Float64(-l)), Float64(expm1(Float64(l + l)) * J), U);
	else
		tmp = Float64(Float64(Float64(J * expm1(Float64(2.0 * l))) / Float64(1.0 + l)) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 500.0], N[(N[Exp[(-l)], $MachinePrecision] * N[(N[(Exp[N[(l + l), $MachinePrecision]] - 1), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(Exp[N[(2.0 * l), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 500

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}}} + U \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)\right) \cdot \frac{1}{e^{\ell}}} + U \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{e^{\ell}}} + U \]

      5. exp-neg N/A

         \[\leadsto \left(J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(\ell\right)}} + U \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(\ell\right)} \cdot \left(J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)\right)} + U \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\mathsf{neg}\left(\ell\right)}, J \cdot \mathsf{expm1}\left(2 \cdot \ell\right), U\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{-\ell}, \mathsf{expm1}\left(\ell + \ell\right) \cdot J, U\right)} \]

   if 500 < l

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
   8. Step-by-step derivation

      1. lower-+.f64 62.4%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \color{blue}{\ell}} + U \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.24:\\ \;\;\;\;\frac{J \cdot \left(2 \cdot \ell\right)}{e^{\ell}} + U\\ \mathbf{else}:\\ \;\;\;\;\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \ell} + U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.24)
   (+ (/ (* J (* 2.0 l)) (exp l)) U)
   (+ (/ (* J (expm1 (* 2.0 l))) (+ 1.0 l)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.24) {
		tmp = ((J * (2.0 * l)) / exp(l)) + U;
	} else {
		tmp = ((J * expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.24) {
		tmp = ((J * (2.0 * l)) / Math.exp(l)) + U;
	} else {
		tmp = ((J * Math.expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.24:
		tmp = ((J * (2.0 * l)) / math.exp(l)) + U
	else:
		tmp = ((J * math.expm1((2.0 * l))) / (1.0 + l)) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.24)
		tmp = Float64(Float64(Float64(J * Float64(2.0 * l)) / exp(l)) + U);
	else
		tmp = Float64(Float64(Float64(J * expm1(Float64(2.0 * l))) / Float64(1.0 + l)) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.24], N[(N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(Exp[N[(2.0 * l), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.23999999999999999

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 62.6%

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

   9. Applied rewrites 62.6%

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

   if -0.23999999999999999 < l

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
   8. Step-by-step derivation

      1. lower-+.f64 62.4%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \color{blue}{\ell}} + U \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \ell} + U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.0)
   (fma (* 2.0 (* J l)) (fma (* K K) -0.125 1.0) U)
   (+ (/ (* J (expm1 (* 2.0 l))) (+ 1.0 l)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.0) {
		tmp = fma((2.0 * (J * l)), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = ((J * expm1((2.0 * l))) / (1.0 + l)) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.0)
		tmp = fma(Float64(2.0 * Float64(J * l)), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = Float64(Float64(Float64(J * expm1(Float64(2.0 * l))) / Float64(1.0 + l)) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.0], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(Exp[N[(2.0 * l), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

   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 48.8%

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

   9. Applied rewrites 48.8%

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

   if -8 < l

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}}} + U \]

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 N/A

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

      3. lower-*.f64 62.5%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}} + U \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}}{e^{\ell}} + U \]

   7. Taylor expanded in l around 0

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
   8. Step-by-step derivation

      1. lower-+.f64 62.4%

         \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1 + \color{blue}{\ell}} + U \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{\color{blue}{1 + \ell}} + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -11.5:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+25}:\\ \;\;\;\;t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell \cdot J\right) \cdot 2 + U\right) \cdot U}{U}\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* J l))))
   (if (<= l -11.5)
     (fma t_0 (fma (* K K) -0.125 1.0) U)
     (if (<= l 1.1e+25) (+ t_0 U) (/ (* (+ (* (* l J) 2.0) U) U) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * l);
	double tmp;
	if (l <= -11.5) {
		tmp = fma(t_0, fma((K * K), -0.125, 1.0), U);
	} else if (l <= 1.1e+25) {
		tmp = t_0 + U;
	} else {
		tmp = ((((l * J) * 2.0) + U) * U) / U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(J * l))
	tmp = 0.0
	if (l <= -11.5)
		tmp = fma(t_0, fma(Float64(K * K), -0.125, 1.0), U);
	elseif (l <= 1.1e+25)
		tmp = Float64(t_0 + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * J) * 2.0) + U) * U) / U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -11.5], N[(t$95$0 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.1e+25], N[(t$95$0 + U), $MachinePrecision], N[(N[(N[(N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision] + U), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -11.5

   1. Initial program 86.7%

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

   2. Taylor expanded in 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 64.8%

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   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 64.8%

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

   6. Applied rewrites 69.7%

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

   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 48.8%

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

   9. Applied rewrites 48.8%

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

   if -11.5 < l < 1.1e25

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 63.4%

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 53.7%

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

   7. Applied rewrites 53.7%

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

   if 1.1e25 < l

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 63.4%

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 53.7%

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

   7. Applied rewrites 53.7%

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

      1. lift-+.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

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 41.9%

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

Alternative 15: 57.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\ell \cdot J\right) \cdot 2\\ \mathbf{if}\;\ell \leq 1.16 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{U}, U, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 + U\right) \cdot U}{U}\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* l J) 2.0)))
   (if (<= l 1.16e+25) (fma (/ t_0 U) U U) (/ (* (+ t_0 U) U) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l * J) * 2.0;
	double tmp;
	if (l <= 1.16e+25) {
		tmp = fma((t_0 / U), U, U);
	} else {
		tmp = ((t_0 + U) * U) / U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l * J) * 2.0)
	tmp = 0.0
	if (l <= 1.16e+25)
		tmp = fma(Float64(t_0 / U), U, U);
	else
		tmp = Float64(Float64(Float64(t_0 + U) * U) / U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[l, 1.16e+25], N[(N[(t$95$0 / U), $MachinePrecision] * U + U), $MachinePrecision], N[(N[(N[(t$95$0 + U), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15999999999999992e25

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 63.4%

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 53.7%

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

   7. Applied rewrites 53.7%

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

      1. lift-+.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

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. lower-fma.f64 57.2%

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

   11. Applied rewrites 57.2%

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

   if 1.15999999999999992e25 < l

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 63.4%

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 53.7%

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

   7. Applied rewrites 53.7%

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

      1. lift-+.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

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 41.9%

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

Alternative 16: 57.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

2. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 63.4%

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

4. Applied rewrites 63.4%

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

5. Taylor expanded in K around 0

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

   1. lower-*.f64 53.7%

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

7. Applied rewrites 53.7%

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

   1. lift-+.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

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

9. Applied rewrites 57.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. distribute-rgt-in N/A

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

    6. *-lft-identity N/A

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

    7. lower-fma.f64 57.2%

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

11. Applied rewrites 57.2%

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

Alternative 17: 53.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

2. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 63.4%

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

4. Applied rewrites 63.4%

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

5. Taylor expanded in K around 0

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

   1. lower-*.f64 53.7%

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

7. Applied rewrites 53.7%

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

Alternative 18: 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 86.7%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2025189 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))