Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.2% → 97.4%

Time: 9.0s

Alternatives: 16

Speedup: 2.6×

• 49.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.2% 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

real(8) function code(j, l, k, u)
    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: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 -0.1)
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+
            0.3333333333333333
            (*
             (* l l)
             (+
              0.016666666666666666
              (* (* l l) 0.0003968253968253968)))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    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 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if (t_1 <= (-0.1d0)) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if t_1 <= -0.1:
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.10000000000000001

   1. Initial program 100.0%

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

   if -0.10000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 78.8%

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

   3. Taylor expanded in l around 0 98.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 98.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 98.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 98.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 98.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   10. Step-by-step derivation

       1. unpow2 98.1%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Applied egg-rr 98.1%

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1:\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;e^{\ell} - t\_1 \leq -\infty:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 - t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))))
   (if (<= (- (exp l) t_1) (- INFINITY))
     (+ U (* t_0 (* J (- 0.3333333333333333 t_1))))
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+
            0.3333333333333333
            (*
             (* l l)
             (+
              0.016666666666666666
              (* (* l l) 0.0003968253968253968)))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double tmp;
	if ((exp(l) - t_1) <= -((double) INFINITY)) {
		tmp = U + (t_0 * (J * (0.3333333333333333 - t_1)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double tmp;
	if ((Math.exp(l) - t_1) <= -Double.POSITIVE_INFINITY) {
		tmp = U + (t_0 * (J * (0.3333333333333333 - t_1)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	tmp = 0
	if (math.exp(l) - t_1) <= -math.inf:
		tmp = U + (t_0 * (J * (0.3333333333333333 - t_1)))
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (Float64(exp(l) - t_1) <= Float64(-Inf))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(0.3333333333333333 - t_1))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	tmp = 0.0;
	if ((exp(l) - t_1) <= -Inf)
		tmp = U + (t_0 * (J * (0.3333333333333333 - t_1)));
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision], (-Infinity)], N[(U + N[(t$95$0 * N[(J * N[(0.3333333333333333 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{0.3333333333333333} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 79.1%

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

   3. Taylor expanded in l around 0 97.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 97.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 97.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   10. Step-by-step derivation

       1. unpow2 97.8%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Applied egg-rr 97.8%

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;U + t\_0 \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -1.9e+40)
     (+ U (* t_0 (* 0.0003968253968253968 (* J (pow l 7.0)))))
     (if (<= l -1.1e+21)
       (+ U (* J (- 0.3333333333333333 (exp (- l)))))
       (+
        U
        (*
         t_0
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (* l l)
             (+
              0.3333333333333333
              (*
               (* l l)
               (+
                0.016666666666666666
                (* (* l l) 0.0003968253968253968))))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -1.9e+40) {
		tmp = U + (t_0 * (0.0003968253968253968 * (J * pow(l, 7.0))));
	} else if (l <= -1.1e+21) {
		tmp = U + (J * (0.3333333333333333 - exp(-l)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-1.9d+40)) then
        tmp = u + (t_0 * (0.0003968253968253968d0 * (j * (l ** 7.0d0))))
    else if (l <= (-1.1d+21)) then
        tmp = u + (j * (0.3333333333333333d0 - exp(-l)))
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -1.9e+40) {
		tmp = U + (t_0 * (0.0003968253968253968 * (J * Math.pow(l, 7.0))));
	} else if (l <= -1.1e+21) {
		tmp = U + (J * (0.3333333333333333 - Math.exp(-l)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -1.9e+40:
		tmp = U + (t_0 * (0.0003968253968253968 * (J * math.pow(l, 7.0))))
	elif l <= -1.1e+21:
		tmp = U + (J * (0.3333333333333333 - math.exp(-l)))
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -1.9e+40)
		tmp = Float64(U + Float64(t_0 * Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0)))));
	elseif (l <= -1.1e+21)
		tmp = Float64(U + Float64(J * Float64(0.3333333333333333 - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -1.9e+40)
		tmp = U + (t_0 * (0.0003968253968253968 * (J * (l ^ 7.0))));
	elseif (l <= -1.1e+21)
		tmp = U + (J * (0.3333333333333333 - exp(-l)));
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.9e+40], N[(U + N[(t$95$0 * N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.1e+21], N[(U + N[(J * N[(0.3333333333333333 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.90000000000000002e40

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.90000000000000002e40 < l < -1.1e21

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 90.0%

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

   5. Applied egg-rr 90.0%

      \[\leadsto J \cdot \left(\color{blue}{0.3333333333333333} - e^{-\ell}\right) + U \]

   if -1.1e21 < l

   1. Initial program 79.3%

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

   3. Taylor expanded in l around 0 97.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 97.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 97.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   10. Step-by-step derivation

       1. unpow2 97.8%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Applied egg-rr 97.8%

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (*
    J
    (*
     l
     (+
      2.0
      (*
       (* l l)
       (+
        0.3333333333333333
        (*
         (* l l)
         (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968)))))))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in l around 0 94.6%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 94.6%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 94.6%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Step-by-step derivation

   1. unpow2 94.6%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Applied egg-rr 94.6%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
8. Step-by-step derivation

   1. unpow2 94.6%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

9. Applied egg-rr 94.6%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
10. Step-by-step derivation

    1. unpow2 94.6%

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

11. Applied egg-rr 94.6%

    \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

12. Final simplification 94.6%

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 58.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+29}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 15200:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+87}:\\ \;\;\;\;4 \cdot {U}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (cos (* K 0.5))))))
   (if (<= l -5.2e+123)
     t_0
     (if (<= l -2.65e+29)
       (* U (- U -4.0))
       (if (<= l 15200.0)
         (fma l (* J 2.0) U)
         (if (<= l 7.5e+87) (* 4.0 (pow U 2.0)) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * cos((K * 0.5)));
	double tmp;
	if (l <= -5.2e+123) {
		tmp = t_0;
	} else if (l <= -2.65e+29) {
		tmp = U * (U - -4.0);
	} else if (l <= 15200.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 7.5e+87) {
		tmp = 4.0 * pow(U, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (l <= -5.2e+123)
		tmp = t_0;
	elseif (l <= -2.65e+29)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 15200.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 7.5e+87)
		tmp = Float64(4.0 * (U ^ 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.2e+123], t$95$0, If[LessEqual[l, -2.65e+29], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 15200.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 7.5e+87], N[(4.0 * N[Power[U, 2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.19999999999999971e123 or 7.50000000000000014e87 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 37.5%

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

   4. Taylor expanded in J around inf 37.7%

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

   6. Simplified 37.7%

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

   if -5.19999999999999971e123 < l < -2.65e29

   1. Initial program 100.0%

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

   4. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -2.65e29 < l < 15200

   1. Initial program 71.0%

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

   3. Taylor expanded in K around 0 68.5%

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

   4. Taylor expanded in l around 0 76.4%

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

      1. associate-*r* 76.4%

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

      2. +-commutative 76.4%

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

      3. *-commutative 76.4%

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

      4. fma-define 76.4%

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

   6. Simplified 76.4%

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

   if 15200 < l < 7.50000000000000014e87

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{\frac{-3}{U}} \]

   6. Applied egg-rr 2.3%

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

      1. flip-+ 0.0%

         \[\leadsto \color{blue}{\frac{U \cdot U - U \cdot U}{U - U}} \]

      2. difference-of-squares 0.0%

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

      3. +-inverses 0.0%

         \[\leadsto \frac{\left(U + U\right) \cdot \color{blue}{0}}{U - U} \]

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

         \[\leadsto \frac{\left(U + U\right) \cdot \log 1}{\color{blue}{0}} \]

      6. metadata-eval 0.0%

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

      7. associate-*r/ 0.0%

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

      8. metadata-eval 0.0%

         \[\leadsto \left(U + U\right) \cdot \frac{\color{blue}{0}}{\log 1} \]

      9. +-inverses 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \left(U + U\right) \cdot \frac{U \cdot U - U \cdot U}{\color{blue}{0}} \]

      11. +-inverses 0.0%

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

      12. flip-+ 36.7%

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

      13. count-2 36.7%

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

      14. count-2 36.7%

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

      15. swap-sqr 36.7%

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

      16. metadata-eval 36.7%

          \[\leadsto \color{blue}{4} \cdot \left(U \cdot U\right) \]

      17. pow2 36.7%

          \[\leadsto 4 \cdot \color{blue}{{U}^{2}} \]

   8. Applied egg-rr 36.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+123}:\\ \;\;\;\;J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+29}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 15200:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+87}:\\ \;\;\;\;4 \cdot {U}^{2}\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.4e+123)
   (* J (* l (cos (* K 0.5))))
   (if (<= l -7.5e+28)
     (* U (- U -4.0))
     (if (<= l 0.056) (fma l (* J 2.0) U) (+ U (* J (+ (exp l) -1.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.4e+123) {
		tmp = J * (l * cos((K * 0.5)));
	} else if (l <= -7.5e+28) {
		tmp = U * (U - -4.0);
	} else if (l <= 0.056) {
		tmp = fma(l, (J * 2.0), U);
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.4e+123)
		tmp = Float64(J * Float64(l * cos(Float64(K * 0.5))));
	elseif (l <= -7.5e+28)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 0.056)
		tmp = fma(l, Float64(J * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.4e+123], N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -7.5e+28], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.056], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.39999999999999984e123

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 40.0%

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

   4. Taylor expanded in J around inf 40.5%

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

   6. Simplified 40.5%

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

   if -4.39999999999999984e123 < l < -7.4999999999999998e28

   1. Initial program 100.0%

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

   4. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -7.4999999999999998e28 < l < 0.0560000000000000012

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0 67.8%

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

   4. Taylor expanded in l around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. +-commutative 77.2%

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

      3. *-commutative 77.2%

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

      4. fma-define 77.2%

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

   6. Simplified 77.2%

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

   if 0.0560000000000000012 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 71.4%

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

   4. Taylor expanded in l around 0 70.2%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.95:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.95)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 0.056)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.95) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 0.056) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-0.95d0)) then
        tmp = u + (j * (1.0d0 - exp(-l)))
    else if (l <= 0.056d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.95) {
		tmp = U + (J * (1.0 - Math.exp(-l)));
	} else if (l <= 0.056) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.95:
		tmp = U + (J * (1.0 - math.exp(-l)))
	elif l <= 0.056:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.95)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 0.056)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -0.95)
		tmp = U + (J * (1.0 - exp(-l)));
	elseif (l <= 0.056)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.95], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.056], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 83.9%

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

   4. Taylor expanded in l around 0 83.9%

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

   if -0.94999999999999996 < l < 0.0560000000000000012

   1. Initial program 68.5%

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

   3. Taylor expanded in l around 0 98.5%

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

   if 0.0560000000000000012 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 71.4%

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

   4. Taylor expanded in l around 0 70.2%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.95:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.95:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.95)
   (+ U (* J (- 1.0 (exp (- l)))))
   (if (<= l 0.056) (fma l (* J 2.0) U) (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.95) {
		tmp = U + (J * (1.0 - exp(-l)));
	} else if (l <= 0.056) {
		tmp = fma(l, (J * 2.0), U);
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.95)
		tmp = Float64(U + Float64(J * Float64(1.0 - exp(Float64(-l)))));
	elseif (l <= 0.056)
		tmp = fma(l, Float64(J * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.95], N[(U + N[(J * N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.056], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 83.9%

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

   4. Taylor expanded in l around 0 83.9%

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

   if -0.94999999999999996 < l < 0.0560000000000000012

   1. Initial program 68.5%

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

   3. Taylor expanded in K around 0 67.2%

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

   4. Taylor expanded in l around 0 81.6%

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

      1. associate-*r* 81.6%

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

      2. +-commutative 81.6%

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

      3. *-commutative 81.6%

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

      4. fma-define 81.6%

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

   6. Simplified 81.6%

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

   if 0.0560000000000000012 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 71.4%

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

   4. Taylor expanded in l around 0 70.2%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.95:\\ \;\;\;\;U + J \cdot \left(1 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.98:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.98)
   (+ U (* J (- 0.3333333333333333 (exp (- l)))))
   (if (<= l 0.056) (fma l (* J 2.0) U) (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.98) {
		tmp = U + (J * (0.3333333333333333 - exp(-l)));
	} else if (l <= 0.056) {
		tmp = fma(l, (J * 2.0), U);
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.98)
		tmp = Float64(U + Float64(J * Float64(0.3333333333333333 - exp(Float64(-l)))));
	elseif (l <= 0.056)
		tmp = fma(l, Float64(J * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.98], N[(U + N[(J * N[(0.3333333333333333 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.056], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 84.8%

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

   5. Applied egg-rr 84.8%

      \[\leadsto J \cdot \left(\color{blue}{0.3333333333333333} - e^{-\ell}\right) + U \]

   if -0.97999999999999998 < l < 0.0560000000000000012

   1. Initial program 68.7%

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

   3. Taylor expanded in K around 0 66.8%

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

   4. Taylor expanded in l around 0 81.1%

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

      1. associate-*r* 81.1%

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

      2. +-commutative 81.1%

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

      3. *-commutative 81.1%

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

      4. fma-define 81.2%

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

   6. Simplified 81.2%

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

   if 0.0560000000000000012 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 71.4%

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

   4. Taylor expanded in l around 0 70.2%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.98:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in K around 0 72.6%

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

4. Taylor expanded in l around 0 50.1%

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

   1. associate-*r* 50.1%

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

   2. +-commutative 50.1%

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

   3. *-commutative 50.1%

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

   4. fma-define 50.1%

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

6. Simplified 50.1%

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

7. Final simplification 50.1%

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

Alternative 11: 43.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.95 \lor \neg \left(\ell \leq 2.1\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.95) (not (<= l 2.1))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.95) || !(l <= 2.1)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.95d0)) .or. (.not. (l <= 2.1d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.95) || !(l <= 2.1)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.95) or not (l <= 2.1):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.95) || !(l <= 2.1))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.95) || ~((l <= 2.1)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.95], N[Not[LessEqual[l, 2.1]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.94999999999999996 or 2.10000000000000009 < 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. Add Preprocessing

   4. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -0.94999999999999996 < l < 2.10000000000000009

   1. Initial program 68.7%

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

   3. Taylor expanded in J around 0 66.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.95 \lor \neg \left(\ell \leq 2.1\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.95 \lor \neg \left(\ell \leq 2.1\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.95) (not (<= l 2.1))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.95) || !(l <= 2.1)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.95d0)) .or. (.not. (l <= 2.1d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.95) || !(l <= 2.1)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.95) or not (l <= 2.1):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.95) || !(l <= 2.1))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.95) || ~((l <= 2.1)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.95], N[Not[LessEqual[l, 2.1]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.94999999999999996 or 2.10000000000000009 < 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. Add Preprocessing

   4. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{U \cdot U} \]

   if -0.94999999999999996 < l < 2.10000000000000009

   1. Initial program 68.7%

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

   3. Taylor expanded in J around 0 66.2%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.95 \lor \neg \left(\ell \leq 2.1\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.8% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in K around 0 72.6%

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

4. Taylor expanded in l around 0 50.1%

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

   1. *-commutative 50.1%

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

   2. associate-*l* 50.1%

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

6. Simplified 50.1%

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

7. Final simplification 50.1%

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

Alternative 14: 37.7% accurate, 312.0× 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

real(8) function code(j, l, k, u)
    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 84.5%

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

3. Taylor expanded in J around 0 34.0%

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

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 16 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 16.0)

C

double code(double J, double l, double K, double U) {
	return 16.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 16.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 16.0;
}

Python

def code(J, l, K, U):
	return 16.0

Julia

function code(J, l, K, U)
	return 16.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 16.0;
end

Wolfram

code[J_, l_, K_, U_] := 16.0

Derivation

1. Initial program 84.5%

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

4. Applied egg-rr 23.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, -4, U\right)} \]
5. Step-by-step derivation

   1. fma-undefine 23.4%

      \[\leadsto \color{blue}{-4 \cdot -4 + U} \]

   2. +-commutative 23.4%

      \[\leadsto \color{blue}{U + -4 \cdot -4} \]

   3. metadata-eval 23.4%

      \[\leadsto U + \color{blue}{16} \]

6. Simplified 23.4%

   \[\leadsto \color{blue}{U + 16} \]

7. Taylor expanded in U around 0 2.8%

   \[\leadsto \color{blue}{16} \]
8. Add Preprocessing

Alternative 16: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 84.5%

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

4. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3, U, -4\right)} \]
5. Step-by-step derivation

   1. fma-undefine 2.7%

      \[\leadsto \color{blue}{-3 \cdot U + -4} \]

   2. *-commutative 2.7%

      \[\leadsto \color{blue}{U \cdot -3} + -4 \]

   3. fma-define 2.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(U, -3, -4\right)} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(U, -3, -4\right)} \]

7. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{-4} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))