Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.7%

Time: 9.4s

Alternatives: 16

Speedup: 2.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 86.0% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ t_2 := e^{\ell} - t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - t\_1\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_2 \leq 10^{-10}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(t\_2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))) (t_2 (- (exp l) t_1)))
   (if (<= t_2 (- INFINITY))
     (+ (* (* J (- 27.0 t_1)) t_0) U)
     (if (<= t_2 1e-10)
       (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
       (+ U (* t_0 (* t_2 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double t_2 = exp(l) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (t_2 <= 1e-10) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (t_0 * (t_2 * J));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double t_2 = Math.exp(l) - t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (t_2 <= 1e-10) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (t_0 * (t_2 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	t_2 = math.exp(l) - t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((J * (27.0 - t_1)) * t_0) + U
	elif t_2 <= 1e-10:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = U + (t_0 * (t_2 * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	t_2 = Float64(exp(l) - t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(J * Float64(27.0 - t_1)) * t_0) + U);
	elseif (t_2 <= 1e-10)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(t_2 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	t_2 = exp(l) - t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	elseif (t_2 <= 1e-10)
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = U + (t_0 * (t_2 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$2, 1e-10], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(t$95$2 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000004e-10

   1. Initial program 73.6%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 1.00000000000000004e-10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - t\_1\right)\\ \mathbf{elif}\;\ell \leq 0.038 \lor \neg \left(\ell \leq 1.3 \cdot 10^{+92}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - t\_1\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))
        (t_1 (exp (- l))))
   (if (<= l -4.6e+104)
     t_0
     (if (<= l -7800.0)
       (+ U (* J (- 3.0 t_1)))
       (if (or (<= l 0.038) (not (<= l 1.3e+92)))
         t_0
         (+ U (* (- (exp l) t_1) J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = exp(-l);
	double tmp;
	if (l <= -4.6e+104) {
		tmp = t_0;
	} else if (l <= -7800.0) {
		tmp = U + (J * (3.0 - t_1));
	} else if ((l <= 0.038) || !(l <= 1.3e+92)) {
		tmp = t_0;
	} else {
		tmp = U + ((exp(l) - t_1) * J);
	}
	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 = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    t_1 = exp(-l)
    if (l <= (-4.6d+104)) then
        tmp = t_0
    else if (l <= (-7800.0d0)) then
        tmp = u + (j * (3.0d0 - t_1))
    else if ((l <= 0.038d0) .or. (.not. (l <= 1.3d+92))) then
        tmp = t_0
    else
        tmp = u + ((exp(l) - t_1) * j)
    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)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = Math.exp(-l);
	double tmp;
	if (l <= -4.6e+104) {
		tmp = t_0;
	} else if (l <= -7800.0) {
		tmp = U + (J * (3.0 - t_1));
	} else if ((l <= 0.038) || !(l <= 1.3e+92)) {
		tmp = t_0;
	} else {
		tmp = U + ((Math.exp(l) - t_1) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	t_1 = math.exp(-l)
	tmp = 0
	if l <= -4.6e+104:
		tmp = t_0
	elif l <= -7800.0:
		tmp = U + (J * (3.0 - t_1))
	elif (l <= 0.038) or not (l <= 1.3e+92):
		tmp = t_0
	else:
		tmp = U + ((math.exp(l) - t_1) * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -4.6e+104)
		tmp = t_0;
	elseif (l <= -7800.0)
		tmp = Float64(U + Float64(J * Float64(3.0 - t_1)));
	elseif ((l <= 0.038) || !(l <= 1.3e+92))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(exp(l) - t_1) * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	t_1 = exp(-l);
	tmp = 0.0;
	if (l <= -4.6e+104)
		tmp = t_0;
	elseif (l <= -7800.0)
		tmp = U + (J * (3.0 - t_1));
	elseif ((l <= 0.038) || ~((l <= 1.3e+92)))
		tmp = t_0;
	else
		tmp = U + ((exp(l) - t_1) * J);
	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[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -4.6e+104], t$95$0, If[LessEqual[l, -7800.0], N[(U + N[(J * N[(3.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 0.038], N[Not[LessEqual[l, 1.3e+92]], $MachinePrecision]], t$95$0, N[(U + N[(N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.59999999999999969e104 or -7800 < l < 0.0379999999999999991 or 1.2999999999999999e92 < l

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -4.59999999999999969e104 < l < -7800

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 90.0%

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

   5. Applied egg-rr 90.0%

      \[\leadsto J \cdot \left(\color{blue}{3} - e^{-\ell}\right) + U \]

   if 0.0379999999999999991 < l < 1.2999999999999999e92

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 94.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+104}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.038 \lor \neg \left(\ell \leq 1.3 \cdot 10^{+92}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - t\_1\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;\ell \leq 0.185 \lor \neg \left(\ell \leq 1.65 \cdot 10^{+92}\right):\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - t\_1\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))))
   (if (<= l -4.0)
     (+ (* (* J (- 27.0 t_1)) t_0) U)
     (if (or (<= l 0.185) (not (<= l 1.65e+92)))
       (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
       (+ U (* (- (exp l) t_1) J))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if ((l <= 0.185) || !(l <= 1.65e+92)) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + ((exp(l) - t_1) * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(-l)
    if (l <= (-4.0d0)) then
        tmp = ((j * (27.0d0 - t_1)) * t_0) + u
    else if ((l <= 0.185d0) .or. (.not. (l <= 1.65d+92))) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = u + ((exp(l) - t_1) * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if ((l <= 0.185) || !(l <= 1.65e+92)) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + ((Math.exp(l) - t_1) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	tmp = 0
	if l <= -4.0:
		tmp = ((J * (27.0 - t_1)) * t_0) + U
	elif (l <= 0.185) or not (l <= 1.65e+92):
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = U + ((math.exp(l) - t_1) * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(Float64(Float64(J * Float64(27.0 - t_1)) * t_0) + U);
	elseif ((l <= 0.185) || !(l <= 1.65e+92))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(U + Float64(Float64(exp(l) - t_1) * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	tmp = 0.0;
	if (l <= -4.0)
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	elseif ((l <= 0.185) || ~((l <= 1.65e+92)))
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = U + ((exp(l) - t_1) * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -4.0], N[(N[(N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[Or[LessEqual[l, 0.185], N[Not[LessEqual[l, 1.65e+92]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4 < l < 0.185 or 1.64999999999999987e92 < l

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 0.185 < l < 1.64999999999999987e92

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 94.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\ell \leq 0.185 \lor \neg \left(\ell \leq 1.65 \cdot 10^{+92}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76 \lor \neg \left(\ell \leq 1.65 \cdot 10^{+92}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))
   (if (<= l -4.6e+104)
     t_0
     (if (<= l -7800.0)
       (+ U (* J (- 3.0 (exp (- l)))))
       (if (or (<= l 0.76) (not (<= l 1.65e+92)))
         t_0
         (+ U (* J (+ (exp l) -1.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double tmp;
	if (l <= -4.6e+104) {
		tmp = t_0;
	} else if (l <= -7800.0) {
		tmp = U + (J * (3.0 - exp(-l)));
	} else if ((l <= 0.76) || !(l <= 1.65e+92)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    if (l <= (-4.6d+104)) then
        tmp = t_0
    else if (l <= (-7800.0d0)) then
        tmp = u + (j * (3.0d0 - exp(-l)))
    else if ((l <= 0.76d0) .or. (.not. (l <= 1.65d+92))) then
        tmp = t_0
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double tmp;
	if (l <= -4.6e+104) {
		tmp = t_0;
	} else if (l <= -7800.0) {
		tmp = U + (J * (3.0 - Math.exp(-l)));
	} else if ((l <= 0.76) || !(l <= 1.65e+92)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	tmp = 0
	if l <= -4.6e+104:
		tmp = t_0
	elif l <= -7800.0:
		tmp = U + (J * (3.0 - math.exp(-l)))
	elif (l <= 0.76) or not (l <= 1.65e+92):
		tmp = t_0
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	tmp = 0.0
	if (l <= -4.6e+104)
		tmp = t_0;
	elseif (l <= -7800.0)
		tmp = Float64(U + Float64(J * Float64(3.0 - exp(Float64(-l)))));
	elseif ((l <= 0.76) || !(l <= 1.65e+92))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	tmp = 0.0;
	if (l <= -4.6e+104)
		tmp = t_0;
	elseif (l <= -7800.0)
		tmp = U + (J * (3.0 - exp(-l)));
	elseif ((l <= 0.76) || ~((l <= 1.65e+92)))
		tmp = t_0;
	else
		tmp = U + (J * (exp(l) + -1.0));
	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[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.6e+104], t$95$0, If[LessEqual[l, -7800.0], N[(U + N[(J * N[(3.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 0.76], N[Not[LessEqual[l, 1.65e+92]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.59999999999999969e104 or -7800 < l < 0.76000000000000001 or 1.64999999999999987e92 < l

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -4.59999999999999969e104 < l < -7800

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 90.0%

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

   5. Applied egg-rr 90.0%

      \[\leadsto J \cdot \left(\color{blue}{3} - e^{-\ell}\right) + U \]

   if 0.76000000000000001 < l < 1.64999999999999987e92

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 94.1%

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

   4. Taylor expanded in l around 0 90.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+104}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76 \lor \neg \left(\ell \leq 1.65 \cdot 10^{+92}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq -650:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6.5e+152)
   (+ U (* (* J (* l 2.0)) (+ 1.0 (* -0.125 (* K K)))))
   (if (<= l -650.0)
     (pow U -3.0)
     (if (<= l 0.76) (+ U (* 2.0 (* l J))) (+ U (* J (+ (exp l) -1.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.5e+152) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else if (l <= -650.0) {
		tmp = pow(U, -3.0);
	} else if (l <= 0.76) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-6.5d+152)) then
        tmp = u + ((j * (l * 2.0d0)) * (1.0d0 + ((-0.125d0) * (k * k))))
    else if (l <= (-650.0d0)) then
        tmp = u ** (-3.0d0)
    else if (l <= 0.76d0) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.5e+152) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else if (l <= -650.0) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 0.76) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6.5e+152:
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))))
	elif l <= -650.0:
		tmp = math.pow(U, -3.0)
	elif l <= 0.76:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6.5e+152)
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	elseif (l <= -650.0)
		tmp = U ^ -3.0;
	elseif (l <= 0.76)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6.5e+152)
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	elseif (l <= -650.0)
		tmp = U ^ -3.0;
	elseif (l <= 0.76)
		tmp = U + (2.0 * (l * J));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6.5e+152], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -650.0], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 0.76], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.4999999999999997e152

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 36.5%

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

      1. *-commutative 36.5%

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

      2. associate-*l* 36.5%

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

   5. Simplified 36.5%

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

   6. Taylor expanded in K around 0 35.5%

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

      1. unpow2 35.5%

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

   8. Applied egg-rr 35.5%

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

   if -6.4999999999999997e152 < l < -650

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 38.9%

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

   if -650 < l < 0.76000000000000001

   1. Initial program 73.6%

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

   3. Taylor expanded in l around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in K around 0 88.0%

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

   if 0.76000000000000001 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 77.8%

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

   4. Taylor expanded in l around 0 76.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq -650:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -7800.0)
   (+ U (* J (- 3.0 (exp (- l)))))
   (if (<= l 0.76)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
     (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7800.0) {
		tmp = U + (J * (3.0 - exp(-l)));
	} else if (l <= 0.76) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-7800.0d0)) then
        tmp = u + (j * (3.0d0 - exp(-l)))
    else if (l <= 0.76d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7800.0) {
		tmp = U + (J * (3.0 - Math.exp(-l)));
	} else if (l <= 0.76) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -7800.0:
		tmp = U + (J * (3.0 - math.exp(-l)))
	elif l <= 0.76:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -7800.0)
		tmp = Float64(U + Float64(J * Float64(3.0 - exp(Float64(-l)))));
	elseif (l <= 0.76)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -7800.0)
		tmp = U + (J * (3.0 - exp(-l)));
	elseif (l <= 0.76)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -7800.0], N[(U + N[(J * N[(3.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.76], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7800

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 78.2%

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

   5. Applied egg-rr 78.2%

      \[\leadsto J \cdot \left(\color{blue}{3} - e^{-\ell}\right) + U \]

   if -7800 < l < 0.76000000000000001

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

   5. Simplified 98.9%

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

   if 0.76000000000000001 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 77.8%

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

   4. Taylor expanded in l around 0 76.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7800:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.1e-5)
   (+ U (* J (- 3.0 (exp (- l)))))
   (if (<= l 0.76) (+ U (* 2.0 (* l J))) (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.1e-5) {
		tmp = U + (J * (3.0 - exp(-l)));
	} else if (l <= 0.76) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-2.1d-5)) then
        tmp = u + (j * (3.0d0 - exp(-l)))
    else if (l <= 0.76d0) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.1e-5) {
		tmp = U + (J * (3.0 - Math.exp(-l)));
	} else if (l <= 0.76) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.1e-5:
		tmp = U + (J * (3.0 - math.exp(-l)))
	elif l <= 0.76:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.1e-5)
		tmp = Float64(U + Float64(J * Float64(3.0 - exp(Float64(-l)))));
	elseif (l <= 0.76)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.1e-5)
		tmp = U + (J * (3.0 - exp(-l)));
	elseif (l <= 0.76)
		tmp = U + (2.0 * (l * J));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.1e-5], N[(U + N[(J * N[(3.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.76], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.09999999999999988e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 75.5%

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

   5. Applied egg-rr 75.7%

      \[\leadsto J \cdot \left(\color{blue}{3} - e^{-\ell}\right) + U \]

   if -2.09999999999999988e-5 < l < 0.76000000000000001

   1. Initial program 73.5%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in K around 0 88.7%

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

   if 0.76000000000000001 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 77.8%

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

   4. Taylor expanded in l around 0 76.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(3 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.76:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -950:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+28}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* J (* l 2.0)) (+ 1.0 (* -0.125 (* K K)))))))
   (if (<= l -6.5e+152)
     t_0
     (if (<= l -950.0)
       (pow U -3.0)
       (if (<= l 2.55e+28) (+ U (* 2.0 (* l J))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	double tmp;
	if (l <= -6.5e+152) {
		tmp = t_0;
	} else if (l <= -950.0) {
		tmp = pow(U, -3.0);
	} else if (l <= 2.55e+28) {
		tmp = U + (2.0 * (l * J));
	} 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)) * (1.0d0 + ((-0.125d0) * (k * k))))
    if (l <= (-6.5d+152)) then
        tmp = t_0
    else if (l <= (-950.0d0)) then
        tmp = u ** (-3.0d0)
    else if (l <= 2.55d+28) then
        tmp = u + (2.0d0 * (l * j))
    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)) * (1.0 + (-0.125 * (K * K))));
	double tmp;
	if (l <= -6.5e+152) {
		tmp = t_0;
	} else if (l <= -950.0) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 2.55e+28) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))))
	tmp = 0
	if l <= -6.5e+152:
		tmp = t_0
	elif l <= -950.0:
		tmp = math.pow(U, -3.0)
	elif l <= 2.55e+28:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))))
	tmp = 0.0
	if (l <= -6.5e+152)
		tmp = t_0;
	elseif (l <= -950.0)
		tmp = U ^ -3.0;
	elseif (l <= 2.55e+28)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	tmp = 0.0;
	if (l <= -6.5e+152)
		tmp = t_0;
	elseif (l <= -950.0)
		tmp = U ^ -3.0;
	elseif (l <= 2.55e+28)
		tmp = U + (2.0 * (l * J));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e+152], t$95$0, If[LessEqual[l, -950.0], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 2.55e+28], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.4999999999999997e152 or 2.5500000000000002e28 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-*l* 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in K around 0 36.4%

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

      1. unpow2 36.4%

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

   8. Applied egg-rr 36.4%

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

   if -6.4999999999999997e152 < l < -950

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 38.9%

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

   if -950 < l < 2.5500000000000002e28

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in K around 0 83.3%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq -950:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+28}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -660:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+28}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -660.0)
   (pow U -4.0)
   (if (<= l 2.55e+28)
     (+ U (* 2.0 (* l J)))
     (+ U (* (* J (* l 2.0)) (+ 1.0 (* -0.125 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -660.0) {
		tmp = pow(U, -4.0);
	} else if (l <= 2.55e+28) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	}
	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 <= (-660.0d0)) then
        tmp = u ** (-4.0d0)
    else if (l <= 2.55d+28) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u + ((j * (l * 2.0d0)) * (1.0d0 + ((-0.125d0) * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -660.0) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 2.55e+28) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -660.0:
		tmp = math.pow(U, -4.0)
	elif l <= 2.55e+28:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -660.0)
		tmp = U ^ -4.0;
	elseif (l <= 2.55e+28)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -660.0)
		tmp = U ^ -4.0;
	elseif (l <= 2.55e+28)
		tmp = U + (2.0 * (l * J));
	else
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -660.0], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 2.55e+28], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -660

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 29.8%

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

   if -660 < l < 2.5500000000000002e28

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in K around 0 83.3%

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

   if 2.5500000000000002e28 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in K around 0 36.7%

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

      1. unpow2 36.7%

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

   8. Applied egg-rr 36.7%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -660:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+28}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-5} \lor \neg \left(\ell \leq 2.55 \cdot 10^{+28}\right):\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.1e-5) (not (<= l 2.55e+28)))
   (+ U (* (* J (* l 2.0)) (+ 1.0 (* -0.125 (* K K)))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e-5) || !(l <= 2.55e+28)) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	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 <= (-2.1d-5)) .or. (.not. (l <= 2.55d+28))) then
        tmp = u + ((j * (l * 2.0d0)) * (1.0d0 + ((-0.125d0) * (k * k))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e-5) || !(l <= 2.55e+28)) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.1e-5) or not (l <= 2.55e+28):
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.1e-5) || !(l <= 2.55e+28))
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.1e-5) || ~((l <= 2.55e+28)))
		tmp = U + ((J * (l * 2.0)) * (1.0 + (-0.125 * (K * K))));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.1e-5], N[Not[LessEqual[l, 2.55e+28]], $MachinePrecision]], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.09999999999999988e-5 or 2.5500000000000002e28 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 27.9%

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

      1. *-commutative 27.9%

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

      2. associate-*l* 27.9%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in K around 0 31.3%

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

      1. unpow2 31.3%

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

   8. Applied egg-rr 31.3%

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

   if -2.09999999999999988e-5 < l < 2.5500000000000002e28

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l* 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in K around 0 83.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-5} \lor \neg \left(\ell \leq 2.55 \cdot 10^{+28}\right):\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.9% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3200000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 3200000000000.0) U (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3200000000000.0) {
		tmp = U;
	} else {
		tmp = U * 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 <= 3200000000000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3200000000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 3200000000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 3200000000000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 3200000000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 3200000000000.0], U, N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.2e12

   1. Initial program 82.2%

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

      1. associate-*l* 82.2%

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

      2. fma-define 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in J around 0 49.5%

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

   if 3.2e12 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 13.8%

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

Alternative 12: 55.0% 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 86.8%

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

3. Taylor expanded in l around 0 63.2%

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

   1. *-commutative 63.2%

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

   2. associate-*l* 63.2%

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

5. Simplified 63.2%

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

6. Taylor expanded in K around 0 53.6%

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

7. Final simplification 53.6%

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

Alternative 13: 37.1% 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 86.8%

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

   1. associate-*l* 86.8%

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

   2. fma-define 86.8%

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

3. Simplified 86.8%

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

5. Taylor expanded in J around 0 37.3%

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

Alternative 14: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 8.0)

C

double code(double J, double l, double K, double U) {
	return 8.0;
}

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 = 8.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 8.0;
}

Python

def code(J, l, K, U):
	return 8.0

Julia

function code(J, l, K, U)
	return 8.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 8.0;
end

Wolfram

code[J_, l_, K_, U_] := 8.0

Derivation

1. Initial program 86.8%

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

4. Applied egg-rr 20.5%

   \[\leadsto \color{blue}{8} + U \]

5. Taylor expanded in U around 0 3.0%

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

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.25)

C

double code(double J, double l, double K, double U) {
	return 0.25;
}

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 = 0.25d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.25;
}

Python

def code(J, l, K, U):
	return 0.25

Julia

function code(J, l, K, U)
	return 0.25
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.25;
end

Wolfram

code[J_, l_, K_, U_] := 0.25

Derivation

1. Initial program 86.8%

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

4. Applied egg-rr 20.6%

   \[\leadsto \color{blue}{0.25} + U \]

5. Taylor expanded in U around 0 3.0%

   \[\leadsto \color{blue}{0.25} \]
6. Add Preprocessing

Alternative 16: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. associate-*l* 86.8%

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

   2. fma-define 86.8%

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

3. Simplified 86.8%

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

6. Applied egg-rr 20.2%

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

   1. +-commutative 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in U around 0 2.6%

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

Reproduce

herbie shell --seed 2024169 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))