Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.9%

Time: 4.1s

Alternatives: 15

Speedup: 1.2×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 86.5% accurate, 1.0× speedup

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \ell + \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* (cos (* 0.5 K)) J) (+ l l) 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)) <= -0.05) {
		tmp = fma((cos((0.5 * K)) * J), (l + l), 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)) <= -0.05)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(l + l), 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], -0.05], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(l + l), $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))) < -0.050000000000000003

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.8

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

   4. Applied rewrites 64.8%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 64.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 3: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* (+ l l) J) (cos (* 0.5 K)) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(((l + l) * J), cos((0.5 * K)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(Float64(l + l) * J), cos(Float64(0.5 * K)), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 64.8

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

   4. Applied rewrites 64.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 64.8

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

      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 64.8

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

      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 64.8

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

      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 64.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 4: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot \sinh \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* (* (fma (* K K) -0.125 1.0) J) (sinh l)) 2.0 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)) <= -0.05) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * J) * sinh(l)), 2.0, 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)) <= -0.05)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * J) * sinh(l)), 2.0, 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], -0.05], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + 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))) < -0.050000000000000003

   1. Initial program 86.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.5

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

   6. Applied rewrites 68.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 68.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.5

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

   8. Applied rewrites 68.5%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 5: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), \left(J + J\right) \cdot \sinh \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (fma -0.125 (* K K) 1.0) (* (+ J J) (sinh l)) 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)) <= -0.05) {
		tmp = fma(fma(-0.125, (K * K), 1.0), ((J + J) * sinh(l)), 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)) <= -0.05)
		tmp = fma(fma(-0.125, Float64(K * K), 1.0), Float64(Float64(J + J) * sinh(l)), 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], -0.05], N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $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))) < -0.050000000000000003

   1. Initial program 86.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.5

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

   6. Applied rewrites 68.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 68.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.5

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

   8. Applied rewrites 68.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-sinh.f64 N/A

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

      8. sinh-undef N/A

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

      9. lower-fma.f64 N/A

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

   10. Applied rewrites 69.1%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 6: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\left(J + J\right) \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (+ (* (+ J J) (* (fma -0.125 (* K K) 1.0) l)) 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)) <= -0.05) {
		tmp = ((J + J) * (fma(-0.125, (K * K), 1.0) * l)) + 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)) <= -0.05)
		tmp = Float64(Float64(Float64(J + J) * Float64(fma(-0.125, Float64(K * K), 1.0) * l)) + 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], -0.05], N[(N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * l), $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))) < -0.050000000000000003

   1. Initial program 86.5%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 49.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt1-in N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-*.f64 49.7

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

   9. Applied rewrites 49.7%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 7: 70.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J + J\right) \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.4e-35)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 2.7e-12)
     (fma (+ J J) l U)
     (+ (* (+ J J) (* (fma -0.125 (* K K) 1.0) l)) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.4e-35) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 2.7e-12) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = ((J + J) * (fma(-0.125, (K * K), 1.0) * l)) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.4e-35)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 2.7e-12)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(Float64(Float64(J + J) * Float64(fma(-0.125, Float64(K * K), 1.0) * l)) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.4e-35], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e-12], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e-35

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   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 56.1%

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

   if -1.4e-35 < l < 2.6999999999999998e-12

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

   if 2.6999999999999998e-12 < l

   1. Initial program 86.5%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 49.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt1-in N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-*.f64 49.7

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

   9. Applied rewrites 49.7%

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

Alternative 8: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(J + J\right) \cdot \left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell\right) + U\\ \mathbf{elif}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(U \cdot \mathsf{fma}\left(2, \frac{\ell}{U}, \frac{1}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (+ (* (+ J J) (* (fma -0.125 (* K K) 1.0) l)) U)
     (if (<= t_0 1e+304)
       (fma (+ J J) l U)
       (* J (* U (fma 2.0 (/ l U) (/ 1.0 J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((J + J) * (fma(-0.125, (K * K), 1.0) * l)) + U;
	} else if (t_0 <= 1e+304) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = J * (U * fma(2.0, (l / U), (1.0 / J)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(J + J) * Float64(fma(-0.125, Float64(K * K), 1.0) * l)) + U);
	elseif (t_0 <= 1e+304)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(J * Float64(U * fma(2.0, Float64(l / U), Float64(1.0 / J))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(J * N[(U * N[(2.0 * N[(l / U), $MachinePrecision] + N[(1.0 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.5%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 49.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt1-in N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-*.f64 49.7

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

   9. Applied rewrites 49.7%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999994e303

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

   if 9.9999999999999994e303 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

   11. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 57.2

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

   13. Applied rewrites 57.2%

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

Alternative 9: 61.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot \ell, 2, U\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(U \cdot \mathsf{fma}\left(2, \frac{\ell}{U}, \frac{1}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma (* (* (fma (* K K) -0.125 1.0) J) l) 2.0 U)
     (if (<= t_0 1e+304)
       (fma (+ J J) l U)
       (* J (* U (fma 2.0 (/ l U) (/ 1.0 J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * J) * l), 2.0, U);
	} else if (t_0 <= 1e+304) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = J * (U * fma(2.0, (l / U), (1.0 / J)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * J) * l), 2.0, U);
	elseif (t_0 <= 1e+304)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(J * Float64(U * fma(2.0, Float64(l / U), Float64(1.0 / J))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * 2.0 + U), $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(J * N[(U * N[(2.0 * N[(l / U), $MachinePrecision] + N[(1.0 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.5

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

   6. Applied rewrites 68.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 68.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.5

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 48.0%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999994e303

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

   if 9.9999999999999994e303 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

   11. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 57.2

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

   13. Applied rewrites 57.2%

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

Alternative 10: 61.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(U \cdot \mathsf{fma}\left(2, \frac{\ell}{U}, \frac{1}{J}\right)\right)\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 900:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* U (fma 2.0 (/ l U) (/ 1.0 J))))))
   (if (<= l -9.5e+38) t_0 (if (<= l 900.0) (fma (+ J J) l U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (U * fma(2.0, (l / U), (1.0 / J)));
	double tmp;
	if (l <= -9.5e+38) {
		tmp = t_0;
	} else if (l <= 900.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(U * fma(2.0, Float64(l / U), Float64(1.0 / J))))
	tmp = 0.0
	if (l <= -9.5e+38)
		tmp = t_0;
	elseif (l <= 900.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(U * N[(2.0 * N[(l / U), $MachinePrecision] + N[(1.0 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.5e+38], t$95$0, If[LessEqual[l, 900.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.4999999999999995e38 or 900 < l

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

   11. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 57.2

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

   13. Applied rewrites 57.2%

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

   if -9.4999999999999995e38 < l < 900

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

Alternative 11: 56.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \mathsf{fma}\left(2, \ell, \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) 1e-14)
   (fma (+ J J) l U)
   (* J (fma 2.0 l (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= 1e-14) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = J * fma(2.0, l, (U / J));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= 1e-14)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(J * fma(2.0, l, Float64(U / J)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], 1e-14], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(J * N[(2.0 * l + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 9.99999999999999999e-15

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

   if 9.99999999999999999e-15 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

Alternative 12: 56.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + \frac{U}{J \cdot \ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) 2e-6)
   (fma (+ J J) l U)
   (* J (* l (+ 2.0 (/ U (* J l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= 2e-6) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = J * (l * (2.0 + (U / (J * l))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= 2e-6)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(U / Float64(J * l)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(U / N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.7%

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

   if 1.99999999999999991e-6 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

   11. Taylor expanded in l around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 39.8

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

   13. Applied rewrites 39.8%

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

Alternative 13: 54.7% accurate, 7.9× speedup

Math

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

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 86.5%

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

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 73.9

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

4. Applied rewrites 73.9%

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{-\ell}\right)} \]
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 80.9

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

6. Applied rewrites 80.9%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 54.7%

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

Alternative 14: 44.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;J \leq -4.55 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;J \leq 5.4 \cdot 10^{+135}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= J -4.55e+198) t_0 (if (<= J 5.4e+135) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (J <= -4.55e+198) {
		tmp = t_0;
	} else if (J <= 5.4e+135) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (2.0d0 * l)
    if (j <= (-4.55d+198)) then
        tmp = t_0
    else if (j <= 5.4d+135) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (J <= -4.55e+198) {
		tmp = t_0;
	} else if (J <= 5.4e+135) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	tmp = 0
	if J <= -4.55e+198:
		tmp = t_0
	elif J <= 5.4e+135:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * l))
	tmp = 0.0
	if (J <= -4.55e+198)
		tmp = t_0;
	elseif (J <= 5.4e+135)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * l);
	tmp = 0.0;
	if (J <= -4.55e+198)
		tmp = t_0;
	elseif (J <= 5.4e+135)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.55e+198], t$95$0, If[LessEqual[J, 5.4e+135], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if J < -4.5499999999999998e198 or 5.3999999999999997e135 < J

   1. Initial program 86.5%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 51.9

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

   10. Applied rewrites 51.9%

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

   11. Taylor expanded in J around inf

       \[\leadsto J \cdot \left(2 \cdot \ell\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 19.6

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

   13. Applied rewrites 19.6%

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

   if -4.5499999999999998e198 < J < 5.3999999999999997e135

   1. Initial program 86.5%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 37.8%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.8% accurate, 68.7× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

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 86.5%

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

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 2025150 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))