Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.9%

Time: 14.4s

Alternatives: 19

Speedup: 1.4×

• 49.6% 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: 86.4% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -0.1 \lor \neg \left(t_0 \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;J \cdot \left(t_0 \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.1) (not (<= t_0 4e-8)))
     (+ (* J (* t_0 (cos (* 0.5 K)))) U)
     (+
      U
      (*
       (cos (/ K 2.0))
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 4e-8)) {
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-0.1d0)) .or. (.not. (t_0 <= 4d-8))) then
        tmp = (j * (t_0 * cos((0.5d0 * k)))) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.016666666666666666d0 * (l ** 5.0d0)) + ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))))
    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 tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 4e-8)) {
		tmp = (J * (t_0 * Math.cos((0.5 * K)))) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.1) or not (t_0 <= 4e-8):
		tmp = (J * (t_0 * math.cos((0.5 * K)))) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.1) || !(t_0 <= 4e-8))
		tmp = Float64(Float64(J * Float64(t_0 * cos(Float64(0.5 * K)))) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -0.1) || ~((t_0 <= 4e-8)))
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	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]}, If[Or[LessEqual[t$95$0, -0.1], N[Not[LessEqual[t$95$0, 4e-8]], $MachinePrecision]], N[(N[(J * N[(t$95$0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $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 or 4.0000000000000001e-8 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 J around 0 100.0%

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

   if -0.10000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.0000000000000001e-8

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.02 \lor \neg \left(t_1 \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;J \cdot \left(t_1 \cdot t_0\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(t_0 \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.02) (not (<= t_1 4e-8)))
     (+ (* J (* t_1 t_0)) U)
     (+ U (* (* l J) (* t_0 2.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -0.02) || !(t_1 <= 4e-8)) {
		tmp = (J * (t_1 * t_0)) + U;
	} else {
		tmp = U + ((l * J) * (t_0 * 2.0));
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((0.5d0 * k))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.02d0)) .or. (.not. (t_1 <= 4d-8))) then
        tmp = (j * (t_1 * t_0)) + u
    else
        tmp = u + ((l * j) * (t_0 * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.02) || !(t_1 <= 4e-8)) {
		tmp = (J * (t_1 * t_0)) + U;
	} else {
		tmp = U + ((l * J) * (t_0 * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -0.02) or not (t_1 <= 4e-8):
		tmp = (J * (t_1 * t_0)) + U
	else:
		tmp = U + ((l * J) * (t_0 * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -0.02) || !(t_1 <= 4e-8))
		tmp = Float64(Float64(J * Float64(t_1 * t_0)) + U);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(t_0 * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((0.5 * K));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -0.02) || ~((t_1 <= 4e-8)))
		tmp = (J * (t_1 * t_0)) + U;
	else
		tmp = U + ((l * J) * (t_0 * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.02], N[Not[LessEqual[t$95$1, 4e-8]], $MachinePrecision]], N[(N[(J * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0200000000000000004 or 4.0000000000000001e-8 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 J around 0 99.9%

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

   if -0.0200000000000000004 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.0000000000000001e-8

   1. Initial program 76.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.02 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -0.1:\\ \;\;\;\;J \cdot \left(t_0 \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right) + \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= t_0 -0.1)
     (+ (* J (* t_0 (cos (* 0.5 K)))) U)
     (+
      U
      (*
       (+
        (* 0.0003968253968253968 (* J (pow l 7.0)))
        (+
         (* 0.016666666666666666 (* J (pow l 5.0)))
         (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J)))))
       (cos (/ K 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	} else {
		tmp = U + (((0.0003968253968253968 * (J * pow(l, 7.0))) + ((0.016666666666666666 * (J * pow(l, 5.0))) + ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J))))) * cos((K / 2.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if (t_0 <= (-0.1d0)) then
        tmp = (j * (t_0 * cos((0.5d0 * k)))) + u
    else
        tmp = u + (((0.0003968253968253968d0 * (j * (l ** 7.0d0))) + ((0.016666666666666666d0 * (j * (l ** 5.0d0))) + ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j))))) * cos((k / 2.0d0)))
    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 tmp;
	if (t_0 <= -0.1) {
		tmp = (J * (t_0 * Math.cos((0.5 * K)))) + U;
	} else {
		tmp = U + (((0.0003968253968253968 * (J * Math.pow(l, 7.0))) + ((0.016666666666666666 * (J * Math.pow(l, 5.0))) + ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J))))) * Math.cos((K / 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if t_0 <= -0.1:
		tmp = (J * (t_0 * math.cos((0.5 * K)))) + U
	else:
		tmp = U + (((0.0003968253968253968 * (J * math.pow(l, 7.0))) + ((0.016666666666666666 * (J * math.pow(l, 5.0))) + ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J))))) * math.cos((K / 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(J * Float64(t_0 * cos(Float64(0.5 * K)))) + U);
	else
		tmp = Float64(U + Float64(Float64(Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0))) + Float64(Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))) + Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J))))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	else
		tmp = U + (((0.0003968253968253968 * (J * (l ^ 7.0))) + ((0.016666666666666666 * (J * (l ^ 5.0))) + ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J))))) * cos((K / 2.0)));
	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]}, If[LessEqual[t$95$0, -0.1], N[(N[(J * N[(t$95$0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $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 99.9%

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

   2. Taylor expanded in J around 0 100.0%

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

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

   1. Initial program 83.4%

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

   2. Taylor expanded in l around 0 97.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 97.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right) + \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1:\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right) + \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 4: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -0.1:\\ \;\;\;\;J \cdot \left(t_0 \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= t_0 -0.1)
     (+ (* J (* t_0 (cos (* 0.5 K)))) U)
     (+
      U
      (*
       (cos (/ K 2.0))
       (*
        J
        (+
         (* 0.0003968253968253968 (pow l 7.0))
         (+
          (* 0.016666666666666666 (pow l 5.0))
          (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * ((0.0003968253968253968 * pow(l, 7.0)) + ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if (t_0 <= (-0.1d0)) then
        tmp = (j * (t_0 * cos((0.5d0 * k)))) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.0003968253968253968d0 * (l ** 7.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))))
    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 tmp;
	if (t_0 <= -0.1) {
		tmp = (J * (t_0 * Math.cos((0.5 * K)))) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.0003968253968253968 * Math.pow(l, 7.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if t_0 <= -0.1:
		tmp = (J * (t_0 * math.cos((0.5 * K)))) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.0003968253968253968 * math.pow(l, 7.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(J * Float64(t_0 * cos(Float64(0.5 * K)))) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.0003968253968253968 * (l ^ 7.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * ((0.0003968253968253968 * (l ^ 7.0)) + ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))))));
	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]}, If[LessEqual[t$95$0, -0.1], N[(N[(J * N[(t$95$0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $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 99.9%

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

   2. Taylor expanded in J around 0 100.0%

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

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

   1. Initial program 83.4%

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

   2. Taylor expanded in l around 0 97.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1:\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-40} \lor \neg \left(t_0 \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;U + t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (or (<= t_0 -2e-40) (not (<= t_0 2e+46)))
     (+ U t_0)
     (+ U (* (* l J) (* (cos (* 0.5 K)) 2.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if ((t_0 <= -2e-40) || !(t_0 <= 2e+46)) {
		tmp = U + t_0;
	} else {
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    if ((t_0 <= (-2d-40)) .or. (.not. (t_0 <= 2d+46))) then
        tmp = u + t_0
    else
        tmp = u + ((l * j) * (cos((0.5d0 * k)) * 2.0d0))
    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)) * J;
	double tmp;
	if ((t_0 <= -2e-40) || !(t_0 <= 2e+46)) {
		tmp = U + t_0;
	} else {
		tmp = U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if (t_0 <= -2e-40) or not (t_0 <= 2e+46):
		tmp = U + t_0
	else:
		tmp = U + ((l * J) * (math.cos((0.5 * K)) * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if ((t_0 <= -2e-40) || !(t_0 <= 2e+46))
		tmp = Float64(U + t_0);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(cos(Float64(0.5 * K)) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if ((t_0 <= -2e-40) || ~((t_0 <= 2e+46)))
		tmp = U + t_0;
	else
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-40], N[Not[LessEqual[t$95$0, 2e+46]], $MachinePrecision]], N[(U + t$95$0), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.9999999999999999e-40 or 2e46 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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 72.4%

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

   if -1.9999999999999999e-40 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2e46

   1. Initial program 76.9%

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

   2. Taylor expanded in l around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. associate-*l* 99.4%

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

      4. *-commutative 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -2 \cdot 10^{-40} \lor \neg \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \end{array} \]

Alternative 6: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.4:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 5.5:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 7.0) (* J 0.0003968253968253968))))))
   (if (<= l -2.15e+49)
     t_1
     (if (<= l -0.4)
       (+ U (* (- (exp l) (exp (- l))) J))
       (if (<= l 5.5)
         (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -2.15e+49) {
		tmp = t_1;
	} else if (l <= -0.4) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else if (l <= 5.5) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = t_1;
	}
	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 = u + (t_0 * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    if (l <= (-2.15d+49)) then
        tmp = t_1
    else if (l <= (-0.4d0)) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else if (l <= 5.5d0) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    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.cos((K / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -2.15e+49) {
		tmp = t_1;
	} else if (l <= -0.4) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else if (l <= 5.5) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 7.0) * (J * 0.0003968253968253968)))
	tmp = 0
	if l <= -2.15e+49:
		tmp = t_1
	elif l <= -0.4:
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	elif l <= 5.5:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968))))
	tmp = 0.0
	if (l <= -2.15e+49)
		tmp = t_1;
	elseif (l <= -0.4)
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	elseif (l <= 5.5)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 7.0) * (J * 0.0003968253968253968)));
	tmp = 0.0;
	if (l <= -2.15e+49)
		tmp = t_1;
	elseif (l <= -0.4)
		tmp = U + ((exp(l) - exp(-l)) * J);
	elseif (l <= 5.5)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	else
		tmp = t_1;
	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[(U + N[(t$95$0 * N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.15e+49], t$95$1, If[LessEqual[l, -0.4], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.15e49 or 5.5 < 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 l around 0 96.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 96.3%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-*l* 96.3%

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

   5. Simplified 96.3%

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

   if -2.15e49 < l < -0.40000000000000002

   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 75.0%

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

   if -0.40000000000000002 < l < 5.5

   1. Initial program 77.8%

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

   2. Taylor expanded in l around 0 98.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq -0.4:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 5.5:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]

Alternative 7: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.0024:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 4.2:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (* (cos (/ K 2.0)) (* (pow l 7.0) (* J 0.0003968253968253968))))))
   (if (<= l -2.15e+49)
     t_0
     (if (<= l -0.0024)
       (+ U (* (- (exp l) (exp (- l))) J))
       (if (<= l 4.2) (+ U (* (* l J) (* (cos (* 0.5 K)) 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -2.15e+49) {
		tmp = t_0;
	} else if (l <= -0.0024) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else if (l <= 4.2) {
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    if (l <= (-2.15d+49)) then
        tmp = t_0
    else if (l <= (-0.0024d0)) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else if (l <= 4.2d0) then
        tmp = u + ((l * j) * (cos((0.5d0 * k)) * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -2.15e+49) {
		tmp = t_0;
	} else if (l <= -0.0024) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else if (l <= 4.2) {
		tmp = U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 7.0) * (J * 0.0003968253968253968)))
	tmp = 0
	if l <= -2.15e+49:
		tmp = t_0
	elif l <= -0.0024:
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	elif l <= 4.2:
		tmp = U + ((l * J) * (math.cos((0.5 * K)) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968))))
	tmp = 0.0
	if (l <= -2.15e+49)
		tmp = t_0;
	elseif (l <= -0.0024)
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	elseif (l <= 4.2)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(cos(Float64(0.5 * K)) * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 7.0) * (J * 0.0003968253968253968)));
	tmp = 0.0;
	if (l <= -2.15e+49)
		tmp = t_0;
	elseif (l <= -0.0024)
		tmp = U + ((exp(l) - exp(-l)) * J);
	elseif (l <= 4.2)
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.15e+49], t$95$0, If[LessEqual[l, -0.0024], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2], N[(U + N[(N[(l * J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.15e49 or 4.20000000000000018 < 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 l around 0 96.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 96.3%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-*l* 96.3%

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

   5. Simplified 96.3%

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

   if -2.15e49 < l < -0.00239999999999999979

   1. Initial program 99.4%

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

   2. Taylor expanded in K around 0 70.0%

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

   if -0.00239999999999999979 < l < 4.20000000000000018

   1. Initial program 77.4%

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

   2. Taylor expanded in l around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0024:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 4.2:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]

Alternative 8: 67.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ t_1 := U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -620000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 550:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (log1p (expm1 U)))
        (t_1 (+ U (* J (+ (* l 2.0) (* (* l (pow K 2.0)) -0.25))))))
   (if (<= l -7.8e+158)
     t_1
     (if (<= l -620000.0)
       t_0
       (if (<= l 550.0)
         (+ U (* (* l J) (* (cos (* 0.5 K)) 2.0)))
         (if (<= l 4e+141) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = log1p(expm1(U));
	double t_1 = U + (J * ((l * 2.0) + ((l * pow(K, 2.0)) * -0.25)));
	double tmp;
	if (l <= -7.8e+158) {
		tmp = t_1;
	} else if (l <= -620000.0) {
		tmp = t_0;
	} else if (l <= 550.0) {
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	} else if (l <= 4e+141) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.log1p(Math.expm1(U));
	double t_1 = U + (J * ((l * 2.0) + ((l * Math.pow(K, 2.0)) * -0.25)));
	double tmp;
	if (l <= -7.8e+158) {
		tmp = t_1;
	} else if (l <= -620000.0) {
		tmp = t_0;
	} else if (l <= 550.0) {
		tmp = U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
	} else if (l <= 4e+141) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.log1p(math.expm1(U))
	t_1 = U + (J * ((l * 2.0) + ((l * math.pow(K, 2.0)) * -0.25)))
	tmp = 0
	if l <= -7.8e+158:
		tmp = t_1
	elif l <= -620000.0:
		tmp = t_0
	elif l <= 550.0:
		tmp = U + ((l * J) * (math.cos((0.5 * K)) * 2.0))
	elif l <= 4e+141:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = log1p(expm1(U))
	t_1 = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(Float64(l * (K ^ 2.0)) * -0.25))))
	tmp = 0.0
	if (l <= -7.8e+158)
		tmp = t_1;
	elseif (l <= -620000.0)
		tmp = t_0;
	elseif (l <= 550.0)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(cos(Float64(0.5 * K)) * 2.0)));
	elseif (l <= 4e+141)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.8e+158], t$95$1, If[LessEqual[l, -620000.0], t$95$0, If[LessEqual[l, 550.0], N[(U + N[(N[(l * J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+141], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.8e158 or 4.00000000000000007e141 < 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 l around 0 41.2%

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

      1. +-commutative 41.2%

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

      2. *-commutative 41.2%

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

      3. associate-*r* 41.2%

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

      4. associate-*l* 41.2%

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

      5. fma-def 41.2%

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

      6. *-commutative 41.2%

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

      7. *-commutative 41.2%

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

   4. Simplified 41.2%

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

   5. Taylor expanded in K around 0 34.2%

      \[\leadsto \color{blue}{U + \left(-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 34.2%

         \[\leadsto U + \color{blue}{\left(2 \cdot \left(J \cdot \ell\right) + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      2. *-commutative 34.2%

         \[\leadsto U + \left(\color{blue}{\left(J \cdot \ell\right) \cdot 2} + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      3. associate-*l* 34.2%

         \[\leadsto U + \left(\color{blue}{J \cdot \left(\ell \cdot 2\right)} + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      4. *-commutative 34.2%

         \[\leadsto U + \left(J \cdot \left(\ell \cdot 2\right) + \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25}\right) \]

      5. associate-*l* 34.2%

         \[\leadsto U + \left(J \cdot \left(\ell \cdot 2\right) + \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot -0.25\right)}\right) \]

      6. distribute-lft-out 52.1%

         \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2 + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right)} \]

      7. *-commutative 52.1%

         \[\leadsto U + J \cdot \left(\color{blue}{2 \cdot \ell} + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right) \]

   7. Simplified 52.1%

      \[\leadsto \color{blue}{U + J \cdot \left(2 \cdot \ell + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right)} \]

   if -7.8e158 < l < -6.2e5 or 550 < l < 4.00000000000000007e141

   1. Initial program 100.0%

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

   3. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)} \]

   if -6.2e5 < l < 550

   1. Initial program 77.9%

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

   2. Taylor expanded in l around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*r* 97.4%

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

      3. associate-*l* 97.4%

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

      4. *-commutative 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+158}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -620000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{elif}\;\ell \leq 550:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 9: 67.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (+ (* l 2.0) (* (* l (pow K 2.0)) -0.25))))))
   (if (<= l -1.9e+58)
     t_0
     (if (<= l -2e+17)
       (pow U -3.0)
       (if (<= l 0.01) (+ U (* (* l J) (* (cos (* 0.5 K)) 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * ((l * 2.0) + ((l * pow(K, 2.0)) * -0.25)));
	double tmp;
	if (l <= -1.9e+58) {
		tmp = t_0;
	} else if (l <= -2e+17) {
		tmp = pow(U, -3.0);
	} else if (l <= 0.01) {
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = u + (j * ((l * 2.0d0) + ((l * (k ** 2.0d0)) * (-0.25d0))))
    if (l <= (-1.9d+58)) then
        tmp = t_0
    else if (l <= (-2d+17)) then
        tmp = u ** (-3.0d0)
    else if (l <= 0.01d0) then
        tmp = u + ((l * j) * (cos((0.5d0 * k)) * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * ((l * 2.0) + ((l * Math.pow(K, 2.0)) * -0.25)));
	double tmp;
	if (l <= -1.9e+58) {
		tmp = t_0;
	} else if (l <= -2e+17) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 0.01) {
		tmp = U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * ((l * 2.0) + ((l * math.pow(K, 2.0)) * -0.25)))
	tmp = 0
	if l <= -1.9e+58:
		tmp = t_0
	elif l <= -2e+17:
		tmp = math.pow(U, -3.0)
	elif l <= 0.01:
		tmp = U + ((l * J) * (math.cos((0.5 * K)) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(Float64(l * (K ^ 2.0)) * -0.25))))
	tmp = 0.0
	if (l <= -1.9e+58)
		tmp = t_0;
	elseif (l <= -2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 0.01)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(cos(Float64(0.5 * K)) * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * ((l * 2.0) + ((l * (K ^ 2.0)) * -0.25)));
	tmp = 0.0;
	if (l <= -1.9e+58)
		tmp = t_0;
	elseif (l <= -2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 0.01)
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+58], t$95$0, If[LessEqual[l, -2e+17], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 0.01], N[(U + N[(N[(l * J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8999999999999999e58 or 0.0100000000000000002 < 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 l around 0 30.2%

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

      1. +-commutative 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-*r* 30.2%

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

      4. associate-*l* 30.2%

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

      5. fma-def 30.2%

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

      6. *-commutative 30.2%

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

      7. *-commutative 30.2%

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

   4. Simplified 30.2%

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

   5. Taylor expanded in K around 0 30.6%

      \[\leadsto \color{blue}{U + \left(-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto U + \color{blue}{\left(2 \cdot \left(J \cdot \ell\right) + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} \]

      2. *-commutative 30.6%

         \[\leadsto U + \left(\color{blue}{\left(J \cdot \ell\right) \cdot 2} + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      3. associate-*l* 30.6%

         \[\leadsto U + \left(\color{blue}{J \cdot \left(\ell \cdot 2\right)} + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right) \]

      4. *-commutative 30.6%

         \[\leadsto U + \left(J \cdot \left(\ell \cdot 2\right) + \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25}\right) \]

      5. associate-*l* 30.6%

         \[\leadsto U + \left(J \cdot \left(\ell \cdot 2\right) + \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot -0.25\right)}\right) \]

      6. distribute-lft-out 40.3%

         \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2 + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right)} \]

      7. *-commutative 40.3%

         \[\leadsto U + J \cdot \left(\color{blue}{2 \cdot \ell} + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right) \]

   7. Simplified 40.3%

      \[\leadsto \color{blue}{U + J \cdot \left(2 \cdot \ell + \left({K}^{2} \cdot \ell\right) \cdot -0.25\right)} \]

   if -1.8999999999999999e58 < l < -2e17

   1. Initial program 100.0%

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

   3. Applied egg-rr 62.0%

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

   if -2e17 < l < 0.0100000000000000002

   1. Initial program 78.0%

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

   2. Taylor expanded in l around 0 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*r* 95.8%

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

      3. associate-*l* 95.8%

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

      4. *-commutative 95.8%

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

   4. Simplified 95.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + \left(\ell \cdot {K}^{2}\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 10: 66.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot \left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2450:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* l (+ J (* -0.125 (* J (pow K 2.0))))))))
   (if (<= l -3.7e+86)
     t_0
     (if (<= l -2e+17)
       (pow U -3.0)
       (if (<= l 2450.0) (+ U (* (* l J) (* (cos (* 0.5 K)) 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * (J + (-0.125 * (J * pow(K, 2.0)))));
	double tmp;
	if (l <= -3.7e+86) {
		tmp = t_0;
	} else if (l <= -2e+17) {
		tmp = pow(U, -3.0);
	} else if (l <= 2450.0) {
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (l * (j + ((-0.125d0) * (j * (k ** 2.0d0)))))
    if (l <= (-3.7d+86)) then
        tmp = t_0
    else if (l <= (-2d+17)) then
        tmp = u ** (-3.0d0)
    else if (l <= 2450.0d0) then
        tmp = u + ((l * j) * (cos((0.5d0 * k)) * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * (J + (-0.125 * (J * Math.pow(K, 2.0)))));
	double tmp;
	if (l <= -3.7e+86) {
		tmp = t_0;
	} else if (l <= -2e+17) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 2450.0) {
		tmp = U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (l * (J + (-0.125 * (J * math.pow(K, 2.0)))))
	tmp = 0
	if l <= -3.7e+86:
		tmp = t_0
	elif l <= -2e+17:
		tmp = math.pow(U, -3.0)
	elif l <= 2450.0:
		tmp = U + ((l * J) * (math.cos((0.5 * K)) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(l * Float64(J + Float64(-0.125 * Float64(J * (K ^ 2.0))))))
	tmp = 0.0
	if (l <= -3.7e+86)
		tmp = t_0;
	elseif (l <= -2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 2450.0)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(cos(Float64(0.5 * K)) * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (l * (J + (-0.125 * (J * (K ^ 2.0)))));
	tmp = 0.0;
	if (l <= -3.7e+86)
		tmp = t_0;
	elseif (l <= -2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 2450.0)
		tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(l * N[(J + N[(-0.125 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.7e+86], t$95$0, If[LessEqual[l, -2e+17], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 2450.0], N[(U + N[(N[(l * J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.69999999999999992e86 or 2450 < 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 l around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-*r* 30.0%

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

      4. associate-*l* 30.0%

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

      5. fma-def 30.0%

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

      6. *-commutative 30.0%

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

      7. *-commutative 30.0%

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

   4. Simplified 30.0%

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

   5. Taylor expanded in l around inf 29.9%

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

      1. associate-*r* 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in K around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate-*r* 28.4%

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

      3. associate-*r* 28.4%

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

      4. distribute-rgt-out 39.0%

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

   10. Simplified 39.0%

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

   if -3.69999999999999992e86 < l < -2e17

   1. Initial program 100.0%

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

   3. Applied egg-rr 47.7%

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

   if -2e17 < l < 2450

   1. Initial program 78.5%

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

   2. Taylor expanded in l around 0 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 94.8%

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

      3. associate-*l* 94.8%

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

      4. *-commutative 94.8%

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

   4. Simplified 94.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2450:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 11: 56.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2450:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+199}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (cos (* 0.5 K)) (* l J)))))
   (if (<= l -2.1e+62)
     t_0
     (if (<= l -2.2e+17)
       (pow U -3.0)
       (if (<= l 2450.0)
         (+ U (* 2.0 (* l J)))
         (if (<= l 1.9e+199) (pow U -3.0) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (cos((0.5 * K)) * (l * J));
	double tmp;
	if (l <= -2.1e+62) {
		tmp = t_0;
	} else if (l <= -2.2e+17) {
		tmp = pow(U, -3.0);
	} else if (l <= 2450.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1.9e+199) {
		tmp = pow(U, -3.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (cos((0.5d0 * k)) * (l * j))
    if (l <= (-2.1d+62)) then
        tmp = t_0
    else if (l <= (-2.2d+17)) then
        tmp = u ** (-3.0d0)
    else if (l <= 2450.0d0) then
        tmp = u + (2.0d0 * (l * j))
    else if (l <= 1.9d+199) then
        tmp = u ** (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (Math.cos((0.5 * K)) * (l * J));
	double tmp;
	if (l <= -2.1e+62) {
		tmp = t_0;
	} else if (l <= -2.2e+17) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 2450.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1.9e+199) {
		tmp = Math.pow(U, -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (math.cos((0.5 * K)) * (l * J))
	tmp = 0
	if l <= -2.1e+62:
		tmp = t_0
	elif l <= -2.2e+17:
		tmp = math.pow(U, -3.0)
	elif l <= 2450.0:
		tmp = U + (2.0 * (l * J))
	elif l <= 1.9e+199:
		tmp = math.pow(U, -3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(cos(Float64(0.5 * K)) * Float64(l * J)))
	tmp = 0.0
	if (l <= -2.1e+62)
		tmp = t_0;
	elseif (l <= -2.2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 2450.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif (l <= 1.9e+199)
		tmp = U ^ -3.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (cos((0.5 * K)) * (l * J));
	tmp = 0.0;
	if (l <= -2.1e+62)
		tmp = t_0;
	elseif (l <= -2.2e+17)
		tmp = U ^ -3.0;
	elseif (l <= 2450.0)
		tmp = U + (2.0 * (l * J));
	elseif (l <= 1.9e+199)
		tmp = U ^ -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.1e+62], t$95$0, If[LessEqual[l, -2.2e+17], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 2450.0], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.9e+199], N[Power[U, -3.0], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.1e62 or 1.9e199 < 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 l around 0 36.5%

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

      1. +-commutative 36.5%

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

      2. *-commutative 36.5%

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

      3. associate-*r* 36.5%

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

      4. associate-*l* 36.5%

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

      5. fma-def 36.5%

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

      6. *-commutative 36.5%

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

      7. *-commutative 36.5%

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

   4. Simplified 36.5%

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

   5. Taylor expanded in l around inf 36.3%

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

      1. associate-*r* 36.3%

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

   7. Simplified 36.3%

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

   if -2.1e62 < l < -2.2e17 or 2450 < l < 1.9e199

   1. Initial program 100.0%

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

   3. Applied egg-rr 39.1%

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

   if -2.2e17 < l < 2450

   1. Initial program 78.5%

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

   2. Taylor expanded in l around 0 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 94.8%

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

      3. associate-*l* 94.8%

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

      4. *-commutative 94.8%

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

   4. Simplified 94.8%

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

   5. Taylor expanded in K around 0 81.0%

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

      1. +-commutative 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2450:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+199}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \]

Alternative 12: 63.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (l * (cos((0.5 * K)) * (J * 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 + (l * (cos((0.5d0 * k)) * (j * 2.0d0)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (l * (Math.cos((0.5 * K)) * (J * 2.0)));
}

Python

def code(J, l, K, U):
	return U + (l * (math.cos((0.5 * K)) * (J * 2.0)))

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (l * (cos((0.5 * K)) * (J * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 87.9%

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

2. Taylor expanded in l around 0 65.1%

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

   1. associate-*r* 65.1%

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

   2. *-commutative 65.1%

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

   3. associate-*l* 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

Alternative 13: 63.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + ((l * J) * (cos((0.5 * K)) * 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 + ((l * j) * (cos((0.5d0 * k)) * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + ((l * J) * (Math.cos((0.5 * K)) * 2.0));
}

Python

def code(J, l, K, U):
	return U + ((l * J) * (math.cos((0.5 * K)) * 2.0))

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + ((l * J) * (cos((0.5 * K)) * 2.0));
end

Wolfram

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

Derivation

1. Initial program 87.9%

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

2. Taylor expanded in l around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 65.1%

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

   3. associate-*l* 65.1%

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

   4. *-commutative 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

Alternative 14: 53.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 220000 \lor \neg \left(\ell \leq 1.9 \cdot 10^{+243}\right):\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.9e+25)
   (pow U -4.0)
   (if (or (<= l 220000.0) (not (<= l 1.9e+243)))
     (+ U (* 2.0 (* l J)))
     (pow U -3.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.9e+25) {
		tmp = pow(U, -4.0);
	} else if ((l <= 220000.0) || !(l <= 1.9e+243)) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = pow(U, -3.0);
	}
	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 <= (-2.9d+25)) then
        tmp = u ** (-4.0d0)
    else if ((l <= 220000.0d0) .or. (.not. (l <= 1.9d+243))) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u ** (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.9e+25) {
		tmp = Math.pow(U, -4.0);
	} else if ((l <= 220000.0) || !(l <= 1.9e+243)) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = Math.pow(U, -3.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.9e+25:
		tmp = math.pow(U, -4.0)
	elif (l <= 220000.0) or not (l <= 1.9e+243):
		tmp = U + (2.0 * (l * J))
	else:
		tmp = math.pow(U, -3.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.9e+25)
		tmp = U ^ -4.0;
	elseif ((l <= 220000.0) || !(l <= 1.9e+243))
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = U ^ -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.9e+25)
		tmp = U ^ -4.0;
	elseif ((l <= 220000.0) || ~((l <= 1.9e+243)))
		tmp = U + (2.0 * (l * J));
	else
		tmp = U ^ -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.9e+25], N[Power[U, -4.0], $MachinePrecision], If[Or[LessEqual[l, 220000.0], N[Not[LessEqual[l, 1.9e+243]], $MachinePrecision]], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.8999999999999999e25

   1. Initial program 100.0%

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

   3. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -2.8999999999999999e25 < l < 2.2e5 or 1.89999999999999999e243 < l

   1. Initial program 80.6%

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

   2. Taylor expanded in l around 0 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*r* 88.9%

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

      3. associate-*l* 88.9%

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

      4. *-commutative 88.9%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in K around 0 75.6%

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

      1. +-commutative 75.6%

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

   7. Simplified 75.6%

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

   if 2.2e5 < l < 1.89999999999999999e243

   1. Initial program 100.0%

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

   3. Applied egg-rr 31.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 220000 \lor \neg \left(\ell \leq 1.9 \cdot 10^{+243}\right):\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \]

Alternative 15: 45.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0024 \lor \neg \left(\ell \leq 0.016\right):\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0024) (not (<= l 0.016))) (* 2.0 (* l J)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0024) || !(l <= 0.016)) {
		tmp = 2.0 * (l * J);
	} 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 <= (-0.0024d0)) .or. (.not. (l <= 0.016d0))) then
        tmp = 2.0d0 * (l * j)
    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 <= -0.0024) || !(l <= 0.016)) {
		tmp = 2.0 * (l * J);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.0024) or not (l <= 0.016):
		tmp = 2.0 * (l * J)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0024) || !(l <= 0.016))
		tmp = Float64(2.0 * Float64(l * J));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.0024) || ~((l <= 0.016)))
		tmp = 2.0 * (l * J);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0024], N[Not[LessEqual[l, 0.016]], $MachinePrecision]], N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.00239999999999999979 or 0.016 < 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 l around 0 26.9%

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

      1. +-commutative 26.9%

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

      2. *-commutative 26.9%

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

      3. associate-*r* 26.9%

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

      4. associate-*l* 26.9%

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

      5. fma-def 26.9%

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

      6. *-commutative 26.9%

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

      7. *-commutative 26.9%

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

   4. Simplified 26.9%

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

   5. Taylor expanded in l around inf 26.7%

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

      1. associate-*r* 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in K around 0 19.1%

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

   if -0.00239999999999999979 < l < 0.016

   1. Initial program 77.1%

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

   2. Taylor expanded in J around 0 76.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0024 \lor \neg \left(\ell \leq 0.016\right):\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 16: 53.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 + (2.0d0 * (l * j))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

2. Taylor expanded in l around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 65.1%

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

   3. associate-*l* 65.1%

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

   4. *-commutative 65.1%

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

4. Simplified 65.1%

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

5. Taylor expanded in K around 0 53.9%

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

   1. +-commutative 53.9%

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

7. Simplified 53.9%

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

8. Final simplification 53.9%

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

Alternative 17: 39.2% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.072:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -0.072) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.072) {
		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 <= (-0.072d0)) 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 <= -0.072) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.072:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.072)
		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 <= -0.072)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.072], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.0719999999999999946

   1. Initial program 99.9%

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

   3. Applied egg-rr 16.0%

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

   if -0.0719999999999999946 < l

   1. Initial program 83.4%

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

   2. Taylor expanded in J around 0 55.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.072:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 18: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

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 = 1.0d0
end function

Java

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

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 87.9%

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

3. Applied egg-rr 3.0%

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

   1. *-inverses 3.0%

      \[\leadsto \color{blue}{1} \]

5. Simplified 3.0%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 3.0%

   \[\leadsto 1 \]

Alternative 19: 36.6% 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 87.9%

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

2. Taylor expanded in J around 0 41.1%

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

3. Final simplification 41.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023319 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))