Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.9%

Time: 4.7s

Alternatives: 12

Speedup: 1.3×

• 50.0% 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: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 2: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 99.9%

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

   7. lift-cos.f64 N/A

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

   8. cos-neg-rev N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 3: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.00022:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell + \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.00022)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 1.35e-12)
     (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 (l <= -0.00022) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 1.35e-12) {
		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 (l <= -0.00022)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 1.35e-12)
		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[l, -0.00022], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e-12], 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 l < -2.20000000000000008e-4

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

   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 55.9%

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

   if -2.20000000000000008e-4 < l < 1.3499999999999999e-12

   1. Initial program 86.8%

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

   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 63.8%

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

   4. Applied rewrites 63.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 63.8%

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

   if 1.3499999999999999e-12 < l

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 4: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.00022:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.00022)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 1.35e-12)
     (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 (l <= -0.00022) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 1.35e-12) {
		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 (l <= -0.00022)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 1.35e-12)
		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[l, -0.00022], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e-12], 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 3 regimes

2. if l < -2.20000000000000008e-4

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

   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 55.9%

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

   if -2.20000000000000008e-4 < l < 1.3499999999999999e-12

   1. Initial program 86.8%

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

   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 63.8%

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

   4. Applied rewrites 63.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 63.8%

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

      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 63.8%

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

      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 63.8%

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

      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 63.8%

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

   if 1.3499999999999999e-12 < l

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right), \sinh \ell \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (fma (fma (* K K) -0.125 1.0) (* (sinh l) (+ J 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)) <= -0.1) {
		tmp = fma(fma((K * K), -0.125, 1.0), (sinh(l) * (J + 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)) <= -0.1)
		tmp = fma(fma(Float64(K * K), -0.125, 1.0), Float64(sinh(l) * Float64(J + 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], -0.1], N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $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))) < -0.10000000000000001

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 69.7%

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

   8. Applied rewrites 69.7%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 6: 80.4% 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 86.8%

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

2. Taylor expanded in K around 0

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

   1. 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 73.6%

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

4. Applied rewrites 73.6%

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

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

Alternative 7: 69.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-35}:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 840:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\ell, J + J, U\right) \cdot U}{U}\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.5e-35)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 840.0) (fma (+ J J) l U) (/ (* (fma l (+ J J) U) U) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5e-35) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 840.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = (fma(l, (J + J), U) * U) / U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.5e-35)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 840.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(Float64(fma(l, Float64(J + J), U) * U) / U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.5e-35], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 840.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(l * N[(J + J), $MachinePrecision] + U), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5000000000000001e-35

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

   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 55.9%

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

   if -4.5000000000000001e-35 < l < 840

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

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

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 53.7%

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

   11. Applied rewrites 53.7%

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

   if 840 < l

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 41.9%

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

   11. Applied rewrites 41.9%

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

Alternative 8: 60.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \frac{J + J}{U}, 1\right) \cdot U\\ \mathbf{elif}\;\ell \leq 840:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\ell, J + J, U\right) \cdot U}{U}\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.5e-23)
   (* (fma l (/ (+ J J) U) 1.0) U)
   (if (<= l 840.0) (fma (+ J J) l U) (/ (* (fma l (+ J J) U) U) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5e-23) {
		tmp = fma(l, ((J + J) / U), 1.0) * U;
	} else if (l <= 840.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = (fma(l, (J + J), U) * U) / U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.5e-23)
		tmp = Float64(fma(l, Float64(Float64(J + J) / U), 1.0) * U);
	elseif (l <= 840.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(Float64(fma(l, Float64(J + J), U) * U) / U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.5e-23], N[(N[(l * N[(N[(J + J), $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision], If[LessEqual[l, 840.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(l * N[(J + J), $MachinePrecision] + U), $MachinePrecision] * U), $MachinePrecision] / U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.49999999999999975e-23

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 55.0%

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

   11. Applied rewrites 55.0%

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

   if -4.49999999999999975e-23 < l < 840

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

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

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 53.7%

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

   11. Applied rewrites 53.7%

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

   if 840 < l

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. add-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 41.9%

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

   11. Applied rewrites 41.9%

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

Alternative 9: 58.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in K around 0

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

   1. 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 73.6%

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

4. Applied rewrites 73.6%

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

   1. lift-+.f64 N/A

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

   2. sum-to-mult N/A

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

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

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

6. Applied rewrites 79.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 57.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. distribute-rgt-in N/A

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

    6. *-lft-identity N/A

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

    7. lower-fma.f64 57.2%

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

    8. lift-/.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. associate-/l* N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 60.5%

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

11. Applied rewrites 60.5%

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

Alternative 10: 55.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \frac{J + J}{U}, 1\right) \cdot U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) -2e-7)
   (* (fma l (/ (+ J J) U) 1.0) U)
   (fma (+ J J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= -2e-7) {
		tmp = fma(l, ((J + J) / U), 1.0) * U;
	} else {
		tmp = fma((J + J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= -2e-7)
		tmp = Float64(fma(l, Float64(Float64(J + J) / U), 1.0) * U);
	else
		tmp = fma(Float64(J + J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], -2e-7], N[(N[(l * N[(N[(J + J), $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1.9999999999999999e-7

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 55.0%

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

   11. Applied rewrites 55.0%

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

   if -1.9999999999999999e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 86.8%

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

   2. Taylor expanded in K around 0

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

      1. 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 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 57.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

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

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 53.7%

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

   11. Applied rewrites 53.7%

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

Alternative 11: 53.7% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in K around 0

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

   1. 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 73.6%

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

4. Applied rewrites 73.6%

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

   1. lift-+.f64 N/A

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

   2. sum-to-mult N/A

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

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

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

6. Applied rewrites 79.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 57.2%

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

    1. lift-*.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. lift-/.f64 N/A

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

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

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

    5. +-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. lower-fma.f64 53.7%

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

11. Applied rewrites 53.7%

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

Alternative 12: 36.8% 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 86.8%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.8%

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

Reproduce

herbie shell --seed 2025188 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))