Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 100.0%

Time: 4.9s

Alternatives: 13

Speedup: 1.2×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. lift-*.f64 N/A

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

   11. cos-neg-rev N/A

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

   12. lift-cos.f64 N/A

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

   13. lower-*.f64 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 3: 88.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.56:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(-0.5 \cdot K\right) \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.56)
     (+ (* (* J (- (exp l) (exp (- l)))) (+ 1.0 (* -0.125 (pow K 2.0)))) U)
     (if (<= t_0 0.02)
       (fma l (* (cos (* -0.5 K)) (+ J J)) U)
       (fma (+ J J) (sinh l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.56) {
		tmp = ((J * (exp(l) - exp(-l))) * (1.0 + (-0.125 * pow(K, 2.0)))) + U;
	} else if (t_0 <= 0.02) {
		tmp = fma(l, (cos((-0.5 * K)) * (J + J)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.56)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))) + U);
	elseif (t_0 <= 0.02)
		tmp = fma(l, Float64(cos(Float64(-0.5 * K)) * Float64(J + J)), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.56], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(l * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.56000000000000005

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

   if -0.56000000000000005 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0200000000000000004

   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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

   8. Applied rewrites 63.5%

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

   if 0.0200000000000000004 < (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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 4: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J + J\right), \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)
     (if (<= t_0 2e+111)
       (fma (* l (+ J J)) (cos (* -0.5 K)) U)
       (fma (+ J J) (sinh l) U)))))

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(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	} else if (t_0 <= 2e+111) {
		tmp = fma((l * (J + J)), cos((-0.5 * K)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	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(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	elseif (t_0 <= 2e+111)
		tmp = fma(Float64(l * Float64(J + 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_] := 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[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e+111], N[(N[(l * N[(J + J), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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 K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.99999999999999991e111

   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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 63.5%

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

   8. Applied rewrites 63.5%

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

   if 1.99999999999999991e111 < (+.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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 5: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(-0.5 \cdot K\right) \cdot \left(J + J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)
     (if (<= t_0 2e+111)
       (fma l (* (cos (* -0.5 K)) (+ J J)) U)
       (fma (+ J J) (sinh l) U)))))

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(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	} else if (t_0 <= 2e+111) {
		tmp = fma(l, (cos((-0.5 * K)) * (J + J)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	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(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	elseif (t_0 <= 2e+111)
		tmp = fma(l, Float64(cos(Float64(-0.5 * K)) * Float64(J + J)), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(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[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e+111], N[(l * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $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 K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.99999999999999991e111

   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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 63.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

   8. Applied rewrites 63.5%

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

   if 1.99999999999999991e111 < (+.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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 6: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \sinh \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (+ J J) (* (fma -0.125 (* K K) 1.0) (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)) <= -0.04) {
		tmp = fma((J + J), (fma(-0.125, (K * K), 1.0) * 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)) <= -0.04)
		tmp = fma(Float64(J + J), Float64(fma(-0.125, Float64(K * K), 1.0) * 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], -0.04], N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $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))) < -0.0400000000000000008

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.2%

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.2%

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

   8. Applied rewrites 69.2%

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

   if -0.0400000000000000008 < (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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 7: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

   if -0.0400000000000000008 < (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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 8: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (+ 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)) <= -0.04) {
		tmp = fma((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)) <= -0.04)
		tmp = fma(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], -0.04], N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -0.0400000000000000008

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.2%

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.2%

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 48.9%

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

   if -0.0400000000000000008 < (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 72.9%

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

   4. Applied rewrites 72.9%

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

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 80.2%

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

   6. Applied rewrites 80.2%

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

Alternative 9: 70.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -11:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.34 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, J \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -11.0)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 1.34e+26)
     (fma 2.0 (* J l) U)
     (fma (+ 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 <= -11.0) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 1.34e+26) {
		tmp = fma(2.0, (J * l), U);
	} else {
		tmp = fma((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 <= -11.0)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 1.34e+26)
		tmp = fma(2.0, Float64(J * l), U);
	else
		tmp = fma(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, -11.0], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.34e+26], N[(2.0 * N[(J * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -11

   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 72.9%

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

   4. Applied rewrites 72.9%

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

   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 55.3%

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

   if -11 < l < 1.34000000000000007e26

   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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 53.4%

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

   9. Applied rewrites 53.4%

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

   if 1.34000000000000007e26 < 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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.2%

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.2%

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 48.9%

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

Alternative 10: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\ell \cdot J\right) \cdot 2}{U}\right) \cdot U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (fma (+ J J) (* (fma -0.125 (* K K) 1.0) l) U)
   (* (+ 1.0 (/ (* (* l J) 2.0) U)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = fma((J + J), (fma(-0.125, (K * K), 1.0) * l), U);
	} else {
		tmp = (1.0 + (((l * J) * 2.0) / U)) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = fma(Float64(J + J), Float64(fma(-0.125, Float64(K * K), 1.0) * l), U);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(l * J) * 2.0) / U)) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(N[(J + J), $MachinePrecision] * N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8%

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 64.8%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.2%

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.2%

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 48.9%

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

   if -0.0400000000000000008 < (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. 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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 53.4%

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

   9. Applied rewrites 53.4%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. lower-*.f64 57.2%

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 57.2%

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

   11. Applied rewrites 57.2%

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

Alternative 11: 57.2% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (* (+ 1.0 (/ (* (* l J) 2.0) U)) U))

C

double code(double J, double l, double K, double U) {
	return (1.0 + (((l * J) * 2.0) / U)) * 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 = (1.0d0 + (((l * j) * 2.0d0) / u)) * u
end function

Java

public static double code(double J, double l, double K, double U) {
	return (1.0 + (((l * J) * 2.0) / U)) * U;
}

Python

def code(J, l, K, U):
	return (1.0 + (((l * J) * 2.0) / U)) * U

Julia

function code(J, l, K, U)
	return Float64(Float64(1.0 + Float64(Float64(Float64(l * J) * 2.0) / U)) * U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = (1.0 + (((l * J) * 2.0) / U)) * U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(1.0 + N[(N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision] / U), $MachinePrecision]), $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. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.2%

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

7. Taylor expanded in l around 0

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

   1. lower-*.f64 53.4%

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

9. Applied rewrites 53.4%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. sum-to-mult N/A

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

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

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

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

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

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

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

    7. *-commutative N/A

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

    8. lower-*.f64 57.2%

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

    9. lift-*.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 57.2%

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

11. Applied rewrites 57.2%

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

Alternative 12: 53.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := N[(2.0 * N[(J * l), $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. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.2%

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

7. Taylor expanded in l around 0

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

   1. lower-*.f64 53.4%

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

9. Applied rewrites 53.4%

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

Alternative 13: 36.3% accurate, 68.7× speedup

Math

\[U \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.3%

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

Reproduce

herbie shell --seed 2025183 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))