Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.9%

Time: 6.1s

Alternatives: 17

Speedup: 1.2×

• 50.6% 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: 86.3% 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, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot 1\\ t_1 := \sin \left(K \cdot -0.5\right) \cdot 0\\ \mathsf{fma}\left(\frac{t\_1 \cdot t\_1 - t\_0 \cdot t\_0}{t\_1 - t\_0} \cdot \left(\sinh \ell \cdot 2\right), J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (let* ((t_0 (* (cos (* 0.5 K)) 1.0)) (t_1 (* (sin (* K -0.5)) 0.0)))
  (fma
   (* (/ (- (* t_1 t_1) (* t_0 t_0)) (- t_1 t_0)) (* (sinh l) 2.0))
   J
   U)))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K)) * 1.0;
	double t_1 = sin((K * -0.5)) * 0.0;
	return fma(((((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0)) * (sinh(l) * 2.0)), J, U);
}

Julia

function code(J, l, K, U)
	t_0 = Float64(cos(Float64(0.5 * K)) * 1.0)
	t_1 = Float64(sin(Float64(K * -0.5)) * 0.0)
	return fma(Float64(Float64(Float64(Float64(t_1 * t_1) - Float64(t_0 * t_0)) / Float64(t_1 - t_0)) * Float64(sinh(l) * 2.0)), J, U)
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]}, N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Initial program 86.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. flip-+ N/A

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

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

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

5. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(\sin \left(K \cdot -0.5\right) \cdot 0\right) \cdot \left(\sin \left(K \cdot -0.5\right) \cdot 0\right) - \left(\cos \left(0.5 \cdot K\right) \cdot 1\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 1\right)}{\sin \left(K \cdot -0.5\right) \cdot 0 - \cos \left(0.5 \cdot K\right) \cdot 1}} \cdot \left(\sinh \ell \cdot 2\right), J, U\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 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 86.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 4: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.852:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0))))
  (if (<= t_0 -0.852)
    (+
     (* (* J (- (exp l) (exp (- l)))) (+ 1.0 (* -0.125 (pow K 2.0))))
     U)
    (if (<= t_0 -0.01)
      (fma (* (cos (* 0.5 K)) (+ l l)) J U)
      (fma (+ J J) (sinh l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.852) {
		tmp = ((J * (exp(l) - exp(-l))) * (1.0 + (-0.125 * pow(K, 2.0)))) + U;
	} else if (t_0 <= -0.01) {
		tmp = fma((cos((0.5 * K)) * (l + l)), J, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.852)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))) + U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(l + l)), J, U);
	else
		tmp = fma(Float64(J + J), sinh(l), 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, -0.852], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.85199999999999998

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.4%

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

   4. Applied rewrites 64.4%

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

   if -0.85199999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.01

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.7%

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

   4. Applied rewrites 64.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 64.7%

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

   if -0.01 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 5: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.852:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1 + -0.125 \cdot {K}^{2}, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0))))
  (if (<= t_0 -0.852)
    (fma (* (+ J J) (sinh l)) (+ 1.0 (* -0.125 (pow K 2.0))) U)
    (if (<= t_0 -0.01)
      (fma (* (cos (* 0.5 K)) (+ l l)) J U)
      (fma (+ J J) (sinh l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.852) {
		tmp = fma(((J + J) * sinh(l)), (1.0 + (-0.125 * pow(K, 2.0))), U);
	} else if (t_0 <= -0.01) {
		tmp = fma((cos((0.5 * K)) * (l + l)), J, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.852)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), Float64(1.0 + Float64(-0.125 * (K ^ 2.0))), U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(l + l)), J, U);
	else
		tmp = fma(Float64(J + J), sinh(l), 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, -0.852], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.85199999999999998

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   8. Applied rewrites 69.0%

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

   if -0.85199999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.01

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.7%

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

   4. Applied rewrites 64.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 64.7%

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

   if -0.01 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 6: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (if (<= (cos (/ K 2.0)) -0.01)
  (fma (* (cos (* 0.5 K)) (+ l l)) J U)
  (fma (+ J J) (sinh l) U)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.01

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.7%

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

   4. Applied rewrites 64.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 64.7%

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

   if -0.01 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 7: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (if (<= (cos (/ K 2.0)) -0.01)
  (fma (* (+ l l) J) (cos (* 0.5 K)) U)
  (fma (+ J J) (sinh l) U)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.01

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.7%

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

   4. Applied rewrites 64.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.7%

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 64.7%

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   6. Applied rewrites 64.7%

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

   if -0.01 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 8: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.65:\\ \;\;\;\;U + J \cdot \left(1 - \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\ell, J, \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.125\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0))))
  (if (<= t_0 -0.65)
    (+ U (* J (- 1.0 (+ 1.0 (* l (- (* 0.5 l) 1.0))))))
    (if (<= t_0 -0.02)
      (+ (* 2.0 (fma l J (* (* (* (* K K) l) J) -0.125))) U)
      (fma (+ J J) (sinh l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.65) {
		tmp = U + (J * (1.0 - (1.0 + (l * ((0.5 * l) - 1.0)))));
	} else if (t_0 <= -0.02) {
		tmp = (2.0 * fma(l, J, ((((K * K) * l) * J) * -0.125))) + U;
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.65)
		tmp = Float64(U + Float64(J * Float64(1.0 - Float64(1.0 + Float64(l * Float64(Float64(0.5 * l) - 1.0))))));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(2.0 * fma(l, J, Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.125))) + U);
	else
		tmp = fma(Float64(J + J), sinh(l), 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, -0.65], N[(U + N[(J * N[(1.0 - N[(1.0 + N[(l * N[(N[(0.5 * l), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(2.0 * N[(l * J + N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.65000000000000002

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in l around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.1%

         \[\leadsto U + J \cdot \left(1 - \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto U + J \cdot \left(1 - \left(1 + \color{blue}{\ell \cdot \left(0.5 \cdot \ell - 1\right)}\right)\right) \]

   if -0.65000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.02

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   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 64.6%

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 45.0%

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

   7. Applied rewrites 45.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 48.0%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 48.0%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.0%

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

   9. Applied rewrites 48.0%

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

   if -0.02 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 9: 84.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(J * N[(1.0 - N[(1.0 + N[(l * N[(N[(0.5 * l), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.01

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in l around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.1%

         \[\leadsto U + J \cdot \left(1 - \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto U + J \cdot \left(1 - \left(1 + \color{blue}{\ell \cdot \left(0.5 \cdot \ell - 1\right)}\right)\right) \]

   if -0.01 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 80.4%

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

   6. Applied rewrites 80.4%

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

Alternative 10: 75.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell \cdot J, J + J\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (if (<= l -1.25e-6)
  (fma (- 1.0 (exp (- l))) J U)
  (+ U (* l (fma (* 0.3333333333333333 l) (* l J) (+ J J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.25e-6) {
		tmp = fma((1.0 - exp(-l)), J, U);
	} else {
		tmp = U + (l * fma((0.3333333333333333 * l), (l * J), (J + J)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.25e-6)
		tmp = fma(Float64(1.0 - exp(Float64(-l))), J, U);
	else
		tmp = Float64(U + Float64(l * fma(Float64(0.3333333333333333 * l), Float64(l * J), Float64(J + J))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.25e-6], N[(N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(l * N[(N[(0.3333333333333333 * l), $MachinePrecision] * N[(l * J), $MachinePrecision] + N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.2500000000000001e-6

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 55.2%

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

   9. Applied rewrites 55.2%

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

   if -1.2500000000000001e-6 < l

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 70.3%

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

   7. Applied rewrites 70.3%

      \[\leadsto U + \ell \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{2}, 2 \cdot J\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.3%

         \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(J \cdot {\ell}^{2}\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 70.3%

          \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left({\ell}^{2} \cdot J\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 70.3%

          \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 70.3%

          \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(2 \cdot J\right)\right) \]

      18. lift-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 70.3%

          \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(J + J\right)\right) \]

   9. Applied rewrites 70.3%

      \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(J + \color{blue}{J}\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sum-to-mult-rev N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 69.6%

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 69.6%

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

   11. Applied rewrites 69.6%

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

Alternative 11: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.017:\\ \;\;\;\;U + J \cdot \left(1 - \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333 \cdot J, J + J\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (if (<= (cos (/ K 2.0)) -0.017)
  (+ U (* J (- 1.0 (+ 1.0 (* l (- (* 0.5 l) 1.0))))))
  (+ U (* l (fma (* l l) (* 0.3333333333333333 J) (+ J J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.017) {
		tmp = U + (J * (1.0 - (1.0 + (l * ((0.5 * l) - 1.0)))));
	} else {
		tmp = U + (l * fma((l * l), (0.3333333333333333 * J), (J + J)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.017)
		tmp = Float64(U + Float64(J * Float64(1.0 - Float64(1.0 + Float64(l * Float64(Float64(0.5 * l) - 1.0))))));
	else
		tmp = Float64(U + Float64(l * fma(Float64(l * l), Float64(0.3333333333333333 * J), Float64(J + J))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.017], N[(U + N[(J * N[(1.0 - N[(1.0 + N[(l * N[(N[(0.5 * l), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 * J), $MachinePrecision] + N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.017000000000000001

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in l around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.1%

         \[\leadsto U + J \cdot \left(1 - \left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto U + J \cdot \left(1 - \left(1 + \color{blue}{\ell \cdot \left(0.5 \cdot \ell - 1\right)}\right)\right) \]

   if -0.017000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   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 73.3%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 70.3%

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

   7. Applied rewrites 70.3%

      \[\leadsto U + \ell \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{2}, 2 \cdot J\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 70.3%

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 70.3%

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

   9. Applied rewrites 70.3%

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

Alternative 12: 70.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

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 73.3%

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

4. Applied rewrites 73.3%

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

5. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 70.3%

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

7. Applied rewrites 70.3%

   \[\leadsto U + \ell \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{2}, 2 \cdot J\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 70.3%

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

   10. lift-*.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 70.3%

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

9. Applied rewrites 70.3%

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

Alternative 13: 69.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

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 73.3%

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

4. Applied rewrites 73.3%

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

5. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 70.3%

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

7. Applied rewrites 70.3%

   \[\leadsto U + \ell \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{2}, 2 \cdot J\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. sum-to-mult N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f64 70.3%

      \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(J \cdot {\ell}^{2}\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 70.3%

       \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left({\ell}^{2} \cdot J\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 70.3%

       \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{2 \cdot J}\right) \cdot \left(2 \cdot J\right)\right) \]

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 70.3%

       \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(2 \cdot J\right)\right) \]

   18. lift-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 70.3%

       \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(J + J\right)\right) \]

9. Applied rewrites 70.3%

   \[\leadsto U + \ell \cdot \left(\left(1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333}{J + J}\right) \cdot \left(J + \color{blue}{J}\right)\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. sum-to-mult-rev N/A

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

    5. +-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. lift-*.f64 N/A

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

    13. associate-*r* N/A

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

    14. lower-fma.f64 N/A

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

    15. lower-*.f64 69.6%

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

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lower-*.f64 69.6%

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

11. Applied rewrites 69.6%

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

Alternative 14: 58.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\ell \cdot J\right) \cdot 2\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{U}, U, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 + U\right) \cdot U}{U}\\ \end{array} \]

FPCore

(FPCore (J l K U)
  :precision binary64
  (let* ((t_0 (* (* l J) 2.0)))
  (if (<= l 4e+49) (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 <= 4e+49) {
		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 <= 4e+49)
		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, 4e+49], 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 < 3.9999999999999998e49

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   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 64.6%

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

   4. Applied rewrites 64.6%

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

   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 54.6%

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

   7. Applied rewrites 54.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
   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 58.0%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. lower-fma.f64 58.0%

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

   11. Applied rewrites 58.0%

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

   if 3.9999999999999998e49 < l

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   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 64.6%

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

   4. Applied rewrites 64.6%

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

   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 54.6%

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

   7. Applied rewrites 54.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
   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 58.0%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 42.8%

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

Alternative 15: 58.0% accurate, 4.6× 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 86.3%

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

2. Taylor expanded in l around 0

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
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 64.6%

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

4. Applied rewrites 64.6%

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

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 54.6%

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

7. Applied rewrites 54.6%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
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 58.0%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. distribute-rgt-in N/A

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

    6. *-lft-identity N/A

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

    7. lower-fma.f64 58.0%

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

11. Applied rewrites 58.0%

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

Alternative 16: 54.6% accurate, 7.6× 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 86.3%

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

2. Taylor expanded in l around 0

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
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 64.6%

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

4. Applied rewrites 64.6%

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

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 54.6%

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

7. Applied rewrites 54.6%

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

Alternative 17: 37.1% accurate, 72.6× 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 86.3%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2025210 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))