Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.0% → 99.9%

Time: 5.3s

Alternatives: 11

Speedup: 1.2×

• 51.0% 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: 87.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.9%

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. cos-neg-rev N/A

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

   13. lift-cos.f64 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 2: 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 87.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

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 3: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ \mathbf{if}\;\ell \leq -150:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(1 - t\_0\right), J, U\right)\\ \mathbf{elif}\;\ell \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(J + J, t\_1 \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+118}:\\ \;\;\;\;U + J \cdot \left(\frac{1}{t\_0} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (* -0.5 K))))
   (if (<= l -150.0)
     (fma (* t_1 (- 1.0 t_0)) J U)
     (if (<= l 0.52)
       (fma (+ J J) (* t_1 l) U)
       (if (<= l 8.5e+118)
         (+ U (* J (- (/ 1.0 t_0) t_0)))
         (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((-0.5 * K));
	double tmp;
	if (l <= -150.0) {
		tmp = fma((t_1 * (1.0 - t_0)), J, U);
	} else if (l <= 0.52) {
		tmp = fma((J + J), (t_1 * l), U);
	} else if (l <= 8.5e+118) {
		tmp = U + (J * ((1.0 / t_0) - t_0));
	} else {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (l <= -150.0)
		tmp = fma(Float64(t_1 * Float64(1.0 - t_0)), J, U);
	elseif (l <= 0.52)
		tmp = fma(Float64(J + J), Float64(t_1 * l), U);
	elseif (l <= 8.5e+118)
		tmp = Float64(U + Float64(J * Float64(Float64(1.0 / t_0) - t_0)));
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -150.0], N[(N[(t$95$1 * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 0.52], N[(N[(J + J), $MachinePrecision] * N[(t$95$1 * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 8.5e+118], N[(U + N[(J * N[(N[(1.0 / t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -150

   1. Initial program 87.0%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 62.9%

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

   if -150 < l < 0.52000000000000002

   1. Initial program 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

   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. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9%

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lift-cos.f64 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 64.6%

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

   if 0.52000000000000002 < l < 8.5000000000000003e118

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

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

      1. 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 74.2%

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

   4. Applied rewrites 74.2%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. lift-sinh.f64 N/A

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

      4. add-flip N/A

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

      5. cosh-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. sinh-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh---cosh-rev N/A

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

      11. exp-neg N/A

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

      12. lift-exp.f64 N/A

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

      13. lower-/.f64 74.2%

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

   6. Applied rewrites 74.2%

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

   if 8.5000000000000003e118 < l

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.9%

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 63.9%

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

   6. Applied rewrites 68.4%

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

Alternative 4: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(-0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.15)
     (fma (+ J J) (* (cos (* -0.5 K)) l) U)
     (if (<= t_0 -0.004)
       (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) 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 <= -0.15) {
		tmp = fma((J + J), (cos((-0.5 * K)) * l), U);
	} else if (t_0 <= -0.004) {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), 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 <= -0.15)
		tmp = fma(Float64(J + J), Float64(cos(Float64(-0.5 * K)) * l), U);
	elseif (t_0 <= -0.004)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), 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, -0.15], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.004], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(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))) < -0.14999999999999999

   1. Initial program 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

   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. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9%

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lift-cos.f64 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 64.6%

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

   if -0.14999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0040000000000000001

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.9%

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 63.9%

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

   6. Applied rewrites 68.4%

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

   if -0.0040000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

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

      1. 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 74.2%

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 81.1%

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

   6. Applied rewrites 81.1%

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

Alternative 5: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.004)
   (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) 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)) <= -0.004) {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), 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)) <= -0.004)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), 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], -0.004], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(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))) < -0.0040000000000000001

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} + U \]
   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 63.9%

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 63.9%

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

   6. Applied rewrites 68.4%

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

   if -0.0040000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

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

      1. 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 74.2%

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 81.1%

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

   6. Applied rewrites 81.1%

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

Alternative 6: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.068:\\ \;\;\;\;U + \frac{J \cdot \left(\ell \cdot \left(2 + 2 \cdot \ell\right)\right)}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.068)
   (+ U (/ (* J (* l (+ 2.0 (* 2.0 l)))) (exp l)))
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.068) {
		tmp = U + ((J * (l * (2.0 + (2.0 * l)))) / exp(l));
	} 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)) <= -0.068)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(2.0 * l)))) / exp(l)));
	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], -0.068], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $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))) < -0.068000000000000005

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. exp-neg N/A

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

      7. lift-exp.f64 N/A

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

      8. sub-to-fraction N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. prod-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-+.f64 75.8%

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

   3. Applied rewrites 75.8%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.4%

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

   9. Applied rewrites 50.4%

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

   if -0.068000000000000005 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   2. Taylor expanded in K around 0

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

      1. 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 74.2%

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 81.1%

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

   6. Applied rewrites 81.1%

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

Alternative 7: 81.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

2. Taylor expanded in K around 0

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

   1. 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 74.2%

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

4. Applied rewrites 74.2%

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
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. count-2 N/A

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

   13. lift-+.f64 N/A

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

   14. lower-fma.f64 81.1%

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

6. Applied rewrites 81.1%

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

Alternative 8: 58.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\ell \cdot J\right) \cdot 2\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(t\_0 + U\right) \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{U}, U, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* l J) 2.0)))
   (if (<= l -6.5e-23) (/ (* (+ t_0 U) U) U) (fma (/ 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 <= -6.5e-23) {
		tmp = ((t_0 + U) * U) / U;
	} else {
		tmp = fma((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 <= -6.5e-23)
		tmp = Float64(Float64(Float64(t_0 + U) * U) / U);
	else
		tmp = fma(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, -6.5e-23], N[(N[(N[(t$95$0 + U), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision], N[(N[(t$95$0 / U), $MachinePrecision] * U + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.5e-23

   1. Initial program 87.0%

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

   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 54.9%

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

   7. Applied rewrites 54.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
   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 58.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 42.4%

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

   if -6.5e-23 < l

   1. Initial program 87.0%

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

   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 54.9%

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

   7. Applied rewrites 54.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
   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 58.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. lower-fma.f64 58.2%

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

   11. Applied rewrites 58.2%

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

Alternative 9: 58.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

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 54.9%

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

7. Applied rewrites 54.9%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) + U \]
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 58.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. distribute-rgt-in N/A

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

    6. *-lft-identity N/A

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

    7. lower-fma.f64 58.2%

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

11. Applied rewrites 58.2%

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

Alternative 10: 54.9% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

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 54.9%

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

7. Applied rewrites 54.9%

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

Alternative 11: 37.8% accurate, 68.7× speedup

Math

\[U \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))