Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.6% → 99.9%

Time: 5.4s

Alternatives: 14

Speedup: 1.3×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

2. Taylor expanded in J around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-undef N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{if}\;\ell \leq -0.098:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.0115:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)))
   (if (<= l -0.098)
     t_0
     (if (<= l 0.0115) (fma (* (+ J J) (cos (* 0.5 K))) l U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	double tmp;
	if (l <= -0.098) {
		tmp = t_0;
	} else if (l <= 0.0115) {
		tmp = fma(((J + J) * cos((0.5 * K))), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U)
	tmp = 0.0
	if (l <= -0.098)
		tmp = t_0;
	elseif (l <= 0.0115)
		tmp = fma(Float64(Float64(J + J) * cos(Float64(0.5 * K))), l, U);
	else
		tmp = t_0;
	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[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -0.098], t$95$0, If[LessEqual[l, 0.0115], N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.098000000000000004 or 0.0115 < l

   1. Initial program 99.9%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   if -0.098000000000000004 < l < 0.0115

   1. Initial program 71.2%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 85.9%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.8

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

   4. Applied rewrites 67.8%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 4: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (+
    (* (* (* (* (* l l) l) J) 0.3333333333333333) (fma (* K K) -0.125 1.0))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = (((((l * l) * l) * J) * 0.3333333333333333) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.3333333333333333) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 85.9%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.8

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right) \cdot \color{blue}{\ell}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right) \cdot \color{blue}{\ell}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(J \cdot {\ell}^{2}\right) \cdot \frac{1}{3} + 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(J \cdot {\ell}^{2}, \frac{1}{3}, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({\ell}^{2} \cdot J, \frac{1}{3}, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\ell}^{2} \cdot J, \frac{1}{3}, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, \frac{1}{3}, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, \frac{1}{3}, 2 \cdot J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      9. count-2-rev N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, \frac{1}{3}, J + J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      10. lift-+.f64 59.9

          \[\leadsto \left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.3333333333333333, J + J\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   7. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.3333333333333333, J + J\right) \cdot \ell\right)} \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      3. *-commutative N/A

         \[\leadsto \left(\left({\ell}^{3} \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({\ell}^{3} \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      5. unpow3 N/A

         \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      6. pow2 N/A

         \[\leadsto \left(\left(\left({\ell}^{2} \cdot \ell\right) \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left({\ell}^{2} \cdot \ell\right) \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      8. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot \frac{1}{3}\right) \cdot \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right) + U \]

      9. lift-*.f64 63.9

         \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   10. Applied rewrites 63.9%

       \[\leadsto \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot \color{blue}{0.3333333333333333}\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 5: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 55.9

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

   7. Applied rewrites 55.9%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 6: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J + J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (+ J J) (* (fma (* K K) -0.125 1.0) l) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((J + J), (fma((K * K), -0.125, 1.0) * l), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(J + J), Float64(fma(Float64(K * K), -0.125, 1.0) * l), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 55.9

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

   7. Applied rewrites 55.9%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 7: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\left(\left(J + J\right) \cdot \ell\right) \cdot \left(\left(K \cdot K\right) \cdot -0.125\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (+ (* (* (+ J J) l) (* (* K K) -0.125)) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = (((J + J) * l) * ((K * K) * -0.125)) + U;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = Float64(Float64(Float64(Float64(J + J) * l) * Float64(Float64(K * K) * -0.125)) + U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 85.9%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.8

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 51.6

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

   10. Applied rewrites 51.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 8: 71.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.16666666666666666\right)\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(J, \frac{2}{U} \cdot \ell, 1\right) \cdot U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (* (* (* l l) l) J) 0.16666666666666666))))
   (if (<= l -1.02e+101)
     t_0
     (if (<= l 6e+29) (* (fma J (* (/ 2.0 U) l) 1.0) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * ((((l * l) * l) * J) * 0.16666666666666666);
	double tmp;
	if (l <= -1.02e+101) {
		tmp = t_0;
	} else if (l <= 6e+29) {
		tmp = fma(J, ((2.0 / U) * l), 1.0) * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.16666666666666666))
	tmp = 0.0
	if (l <= -1.02e+101)
		tmp = t_0;
	elseif (l <= 6e+29)
		tmp = Float64(fma(J, Float64(Float64(2.0 / U) * l), 1.0) * U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.02e+101], t$95$0, If[LessEqual[l, 6e+29], N[(N[(J * N[(N[(2.0 / U), $MachinePrecision] * l), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.02000000000000002e101 or 5.9999999999999998e29 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. sinh-undef-rev N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(J + \frac{1}{6} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \ell\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(J + \frac{1}{6} \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \ell\right) \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{6} \cdot \left(J \cdot {\ell}^{2}\right) + J\right) \cdot \ell\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\frac{1}{6}, J \cdot {\ell}^{2}, J\right) \cdot \ell\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\frac{1}{6}, {\ell}^{2} \cdot J, J\right) \cdot \ell\right) \]

      6. pow2 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\frac{1}{6}, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\frac{1}{6}, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \]

      8. lift-*.f64 61.7

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \]

   10. Applied rewrites 61.7%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \]

   11. Taylor expanded in l around inf

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

       1. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{6}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \frac{1}{6}\right) \]

       3. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \frac{1}{6}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \frac{1}{6}\right) \]

       5. unpow3 N/A

          \[\leadsto 2 \cdot \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot \frac{1}{6}\right) \]

       6. pow2 N/A

          \[\leadsto 2 \cdot \left(\left(\left({\ell}^{2} \cdot \ell\right) \cdot J\right) \cdot \frac{1}{6}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(\left({\ell}^{2} \cdot \ell\right) \cdot J\right) \cdot \frac{1}{6}\right) \]

       8. pow2 N/A

          \[\leadsto 2 \cdot \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot \frac{1}{6}\right) \]

       9. lift-*.f64 66.7

          \[\leadsto 2 \cdot \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.16666666666666666\right) \]

   13. Applied rewrites 66.7%

       \[\leadsto 2 \cdot \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.16666666666666666\right) \]

   if -1.02000000000000002e101 < l < 5.9999999999999998e29

   1. Initial program 76.3%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. sinh-undef-rev N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sinh.f64 82.0

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

   7. Applied rewrites 82.0%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \left(\frac{1}{3} \cdot \frac{{\ell}^{2}}{U} + 2 \cdot \frac{1}{U}\right), 1\right) \cdot U \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 74.6

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

   10. Applied rewrites 74.6%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 72.6%

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

Alternative 9: 70.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sinh.f64 80.3

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 71.4

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

7. Applied rewrites 71.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right) \]
8. Add Preprocessing

Alternative 10: 60.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \frac{2}{U} \cdot \ell, 1\right) \cdot U \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sinh.f64 80.3

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in U around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. rec-exp N/A

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

   7. lower-/.f64 N/A

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

   8. sinh-undef-rev N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sinh.f64 79.0

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

7. Applied rewrites 79.0%

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

8. Taylor expanded in l around 0

   \[\leadsto \mathsf{fma}\left(J, \ell \cdot \left(\frac{1}{3} \cdot \frac{{\ell}^{2}}{U} + 2 \cdot \frac{1}{U}\right), 1\right) \cdot U \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-add-rev N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 71.8

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

10. Applied rewrites 71.8%

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

11. Taylor expanded in l around 0

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

13. Applied rewrites 60.1%

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

Alternative 11: 60.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, \frac{\ell + \ell}{U}, 1\right) \cdot U \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sinh.f64 80.3

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in U around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. rec-exp N/A

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

   7. lower-/.f64 N/A

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

   8. sinh-undef-rev N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sinh.f64 79.0

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

7. Applied rewrites 79.0%

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

8. Taylor expanded in l around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 60.1

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

10. Applied rewrites 60.1%

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

Alternative 12: 53.8% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

2. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 63.8

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

4. Applied rewrites 63.8%

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

5. Taylor expanded in K around 0

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

7. Applied rewrites 53.8%

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

Alternative 13: 45.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \left(J + J\right) \cdot \ell\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* (+ J J) l)))
   (if (<= t_0 -5e-7) t_1 (if (<= t_0 1e-6) U t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = (J + J) * l;
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_1;
	} else if (t_0 <= 1e-6) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = (j + j) * l
    if (t_0 <= (-5d-7)) then
        tmp = t_1
    else if (t_0 <= 1d-6) then
        tmp = u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double t_1 = (J + J) * l;
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_1;
	} else if (t_0 <= 1e-6) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = (J + J) * l
	tmp = 0
	if t_0 <= -5e-7:
		tmp = t_1
	elif t_0 <= 1e-6:
		tmp = U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(Float64(J + J) * l)
	tmp = 0.0
	if (t_0 <= -5e-7)
		tmp = t_1;
	elseif (t_0 <= 1e-6)
		tmp = U;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = (J + J) * l;
	tmp = 0.0;
	if (t_0 <= -5e-7)
		tmp = t_1;
	elseif (t_0 <= 1e-6)
		tmp = U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-7], t$95$1, If[LessEqual[t$95$0, 1e-6], U, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -4.99999999999999977e-7 or 9.99999999999999955e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.7%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sinh.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. sinh-undef-rev N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 73.4

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

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 21.6%

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

   11. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. count-2-rev N/A

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

       4. lift-+.f64 21.6

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

   13. Applied rewrites 21.6%

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

   if -4.99999999999999977e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 9.99999999999999955e-7

   1. Initial program 70.9%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 70.3%

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

Alternative 14: 36.1% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2025122 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))