Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.4%

Time: 10.1s

Alternatives: 13

Speedup: 2.6×

• 50.0% 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: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ t_2 := e^{\ell} - t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - t\_1\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(t\_2 \cdot J\right)\\ \end{array} \end{array} \]

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 5e-14)
       (+ U (* t_0 (* J (* l 2.0))))
       (+ 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 <= 5e-14) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} 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 <= 5e-14) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} 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 <= 5e-14:
		tmp = U + (t_0 * (J * (l * 2.0)))
	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 <= 5e-14)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	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 <= 5e-14)
		tmp = U + (t_0 * (J * (l * 2.0)));
	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, 5e-14], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $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))) < 5.0000000000000002e-14

   1. Initial program 71.6%

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

   3. Taylor expanded in l around 0 99.9%

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

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

   if 5.0000000000000002e-14 < (-.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;\ell \leq 160 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+88}\right):\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -4.0)
     (+ (* (* J (- 27.0 (exp (- l)))) t_0) U)
     (if (or (<= l 160.0) (not (<= l 5.8e+88)))
       (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
       (+ U (* J (- (exp l) 1.5)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - exp(-l))) * t_0) + U;
	} else if ((l <= 160.0) || !(l <= 5.8e+88)) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-4.0d0)) then
        tmp = ((j * (27.0d0 - exp(-l))) * t_0) + u
    else if ((l <= 160.0d0) .or. (.not. (l <= 5.8d+88))) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - Math.exp(-l))) * t_0) + U;
	} else if ((l <= 160.0) || !(l <= 5.8e+88)) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -4.0:
		tmp = ((J * (27.0 - math.exp(-l))) * t_0) + U
	elif (l <= 160.0) or not (l <= 5.8e+88):
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(Float64(Float64(J * Float64(27.0 - exp(Float64(-l)))) * t_0) + U);
	elseif ((l <= 160.0) || !(l <= 5.8e+88))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -4.0)
		tmp = ((J * (27.0 - exp(-l))) * t_0) + U;
	elseif ((l <= 160.0) || ~((l <= 5.8e+88)))
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = U + (J * (exp(l) - 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.0], N[(N[(N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[Or[LessEqual[l, 160.0], N[Not[LessEqual[l, 5.8e+88]], $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[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $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 < 160 or 5.7999999999999999e88 < l

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0 99.4%

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

      1. unpow2 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 160 < l < 5.7999999999999999e88

   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 82.4%

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

   5. Applied egg-rr 82.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\ell \leq 160 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+88}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -7.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+88}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

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 -2.7e+130)
     t_0
     (if (<= l -7.5)
       (+ (* J (- 27.0 (exp (- l)))) U)
       (if (or (<= l 160.0) (not (<= l 5.8e+88)))
         t_0
         (+ U (* J (- (exp l) 1.5))))))))

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 <= -2.7e+130) {
		tmp = t_0;
	} else if (l <= -7.5) {
		tmp = (J * (27.0 - exp(-l))) + U;
	} else if ((l <= 160.0) || !(l <= 5.8e+88)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	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 <= (-2.7d+130)) then
        tmp = t_0
    else if (l <= (-7.5d0)) then
        tmp = (j * (27.0d0 - exp(-l))) + u
    else if ((l <= 160.0d0) .or. (.not. (l <= 5.8d+88))) then
        tmp = t_0
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    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 <= -2.7e+130) {
		tmp = t_0;
	} else if (l <= -7.5) {
		tmp = (J * (27.0 - Math.exp(-l))) + U;
	} else if ((l <= 160.0) || !(l <= 5.8e+88)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	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 <= -2.7e+130:
		tmp = t_0
	elif l <= -7.5:
		tmp = (J * (27.0 - math.exp(-l))) + U
	elif (l <= 160.0) or not (l <= 5.8e+88):
		tmp = t_0
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	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 <= -2.7e+130)
		tmp = t_0;
	elseif (l <= -7.5)
		tmp = Float64(Float64(J * Float64(27.0 - exp(Float64(-l)))) + U);
	elseif ((l <= 160.0) || !(l <= 5.8e+88))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	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 <= -2.7e+130)
		tmp = t_0;
	elseif (l <= -7.5)
		tmp = (J * (27.0 - exp(-l))) + U;
	elseif ((l <= 160.0) || ~((l <= 5.8e+88)))
		tmp = t_0;
	else
		tmp = U + (J * (exp(l) - 1.5));
	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, -2.7e+130], t$95$0, If[LessEqual[l, -7.5], N[(N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[Or[LessEqual[l, 160.0], N[Not[LessEqual[l, 5.8e+88]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6999999999999998e130 or -7.5 < l < 160 or 5.7999999999999999e88 < l

   1. Initial program 83.3%

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

   3. Taylor expanded in l around 0 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 -2.6999999999999998e130 < l < -7.5

   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 75.0%

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

   5. Applied egg-rr 75.0%

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

   if 160 < l < 5.7999999999999999e88

   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 82.4%

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

   5. Applied egg-rr 82.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -7.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+88}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;\frac{4096 - {U}^{3}}{256}\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l 2.0)))))
   (if (<= l -2.45e+203)
     t_0
     (if (<= l -820.0)
       (/ (- 4096.0 (pow U 3.0)) 256.0)
       (if (<= l 160.0) t_0 (+ U (* J (- (exp l) 1.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * 2.0));
	double tmp;
	if (l <= -2.45e+203) {
		tmp = t_0;
	} else if (l <= -820.0) {
		tmp = (4096.0 - pow(U, 3.0)) / 256.0;
	} else if (l <= 160.0) {
		tmp = t_0;
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	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))
    if (l <= (-2.45d+203)) then
        tmp = t_0
    else if (l <= (-820.0d0)) then
        tmp = (4096.0d0 - (u ** 3.0d0)) / 256.0d0
    else if (l <= 160.0d0) then
        tmp = t_0
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    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));
	double tmp;
	if (l <= -2.45e+203) {
		tmp = t_0;
	} else if (l <= -820.0) {
		tmp = (4096.0 - Math.pow(U, 3.0)) / 256.0;
	} else if (l <= 160.0) {
		tmp = t_0;
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * 2.0))
	tmp = 0
	if l <= -2.45e+203:
		tmp = t_0
	elif l <= -820.0:
		tmp = (4096.0 - math.pow(U, 3.0)) / 256.0
	elif l <= 160.0:
		tmp = t_0
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * 2.0)))
	tmp = 0.0
	if (l <= -2.45e+203)
		tmp = t_0;
	elseif (l <= -820.0)
		tmp = Float64(Float64(4096.0 - (U ^ 3.0)) / 256.0);
	elseif (l <= 160.0)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * 2.0));
	tmp = 0.0;
	if (l <= -2.45e+203)
		tmp = t_0;
	elseif (l <= -820.0)
		tmp = (4096.0 - (U ^ 3.0)) / 256.0;
	elseif (l <= 160.0)
		tmp = t_0;
	else
		tmp = U + (J * (exp(l) - 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.45e+203], t$95$0, If[LessEqual[l, -820.0], N[(N[(4096.0 - N[Power[U, 3.0], $MachinePrecision]), $MachinePrecision] / 256.0), $MachinePrecision], If[LessEqual[l, 160.0], t$95$0, N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.4499999999999999e203 or -820 < l < 160

   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 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*r* 94.6%

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

      3. associate-*l* 94.6%

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

      4. *-commutative 94.6%

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

      5. *-commutative 94.6%

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

      6. *-commutative 94.6%

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

      7. associate-*l* 94.6%

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

   5. Simplified 94.6%

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

   6. Taylor expanded in K around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r* 82.4%

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

      3. *-commutative 82.4%

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

   8. Simplified 82.4%

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

   if -2.4499999999999999e203 < l < -820

   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 3.3%

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

      1. fmm-undef 3.3%

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

      2. metadata-eval 3.3%

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

   6. Simplified 3.3%

      \[\leadsto \color{blue}{16 - U} \]
   7. Step-by-step derivation

      1. flip3-- 5.5%

         \[\leadsto \color{blue}{\frac{{16}^{3} - {U}^{3}}{16 \cdot 16 + \left(U \cdot U + 16 \cdot U\right)}} \]

      2. metadata-eval 5.5%

         \[\leadsto \frac{\color{blue}{4096} - {U}^{3}}{16 \cdot 16 + \left(U \cdot U + 16 \cdot U\right)} \]

      3. metadata-eval 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{\color{blue}{256} + \left(U \cdot U + 16 \cdot U\right)} \]

      4. distribute-rgt-out 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{256 + \color{blue}{U \cdot \left(U + 16\right)}} \]

      5. +-commutative 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{256 + U \cdot \color{blue}{\left(16 + U\right)}} \]

   8. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{\frac{4096 - {U}^{3}}{256 + U \cdot \left(16 + U\right)}} \]

   9. Taylor expanded in U around 0 38.1%

      \[\leadsto \frac{4096 - {U}^{3}}{\color{blue}{256}} \]

   if 160 < 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 79.4%

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

   5. Applied egg-rr 79.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+203}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;\frac{4096 - {U}^{3}}{256}\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.5)
   (+ (* J (- 27.0 (exp (- l)))) U)
   (if (<= l 160.0)
     (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
     (+ U (* J (- (exp l) 1.5))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5) {
		tmp = (J * (27.0 - exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	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 <= (-4.5d0)) then
        tmp = (j * (27.0d0 - exp(-l))) + u
    else if (l <= 160.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5) {
		tmp = (J * (27.0 - Math.exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.5:
		tmp = (J * (27.0 - math.exp(-l))) + U
	elif l <= 160.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.5)
		tmp = Float64(Float64(J * Float64(27.0 - exp(Float64(-l)))) + U);
	elseif (l <= 160.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.5)
		tmp = (J * (27.0 - exp(-l))) + U;
	elseif (l <= 160.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = U + (J * (exp(l) - 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.5], N[(N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 160.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5

   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 80.3%

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

   5. Applied egg-rr 80.3%

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

   if -4.5 < l < 160

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   5. Simplified 99.2%

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

   if 160 < 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 79.4%

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

   5. Applied egg-rr 79.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.0)
   (+ (* J (- 27.0 (exp (- l)))) U)
   (if (<= l 160.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* J (- (exp l) 1.5))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.0) {
		tmp = (J * (27.0 - exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	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 <= (-4.0d0)) then
        tmp = (j * (27.0d0 - exp(-l))) + u
    else if (l <= 160.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.0) {
		tmp = (J * (27.0 - Math.exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.0:
		tmp = (J * (27.0 - math.exp(-l))) + U
	elif l <= 160.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(Float64(J * Float64(27.0 - exp(Float64(-l)))) + U);
	elseif (l <= 160.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.0)
		tmp = (J * (27.0 - exp(-l))) + U;
	elseif (l <= 160.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * (exp(l) - 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.0], N[(N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 160.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $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

   3. Taylor expanded in K around 0 80.3%

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

   5. Applied egg-rr 80.3%

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

   if -4 < l < 160

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 99.2%

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

   if 160 < 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 79.4%

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

   5. Applied egg-rr 79.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.5)
   (+ (* J (- 27.0 (exp (- l)))) U)
   (if (<= l 160.0) (+ U (* J (* l 2.0))) (+ U (* J (- (exp l) 1.5))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.5) {
		tmp = (J * (27.0 - exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (exp(l) - 1.5));
	}
	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 <= (-3.5d0)) then
        tmp = (j * (27.0d0 - exp(-l))) + u
    else if (l <= 160.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = u + (j * (exp(l) - 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.5) {
		tmp = (J * (27.0 - Math.exp(-l))) + U;
	} else if (l <= 160.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (Math.exp(l) - 1.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.5:
		tmp = (J * (27.0 - math.exp(-l))) + U
	elif l <= 160.0:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = U + (J * (math.exp(l) - 1.5))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.5)
		tmp = Float64(Float64(J * Float64(27.0 - exp(Float64(-l)))) + U);
	elseif (l <= 160.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.5)
		tmp = (J * (27.0 - exp(-l))) + U;
	elseif (l <= 160.0)
		tmp = U + (J * (l * 2.0));
	else
		tmp = U + (J * (exp(l) - 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.5], N[(N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 160.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5

   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 80.3%

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

   5. Applied egg-rr 80.3%

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

   if -3.5 < l < 160

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*l* 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in K around 0 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.4%

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

      3. *-commutative 86.4%

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

   8. Simplified 86.4%

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

   if 160 < 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 79.4%

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

   5. Applied egg-rr 79.4%

      \[\leadsto J \cdot \left(e^{\ell} - \color{blue}{1.5}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5:\\ \;\;\;\;J \cdot \left(27 - e^{-\ell}\right) + U\\ \mathbf{elif}\;\ell \leq 160:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+202} \lor \neg \left(\ell \leq -800\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4096 - {U}^{3}}{256}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.05e+202) (not (<= l -800.0)))
   (+ U (* J (* l 2.0)))
   (/ (- 4096.0 (pow U 3.0)) 256.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+202) || !(l <= -800.0)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (4096.0 - pow(U, 3.0)) / 256.0;
	}
	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 <= (-1.05d+202)) .or. (.not. (l <= (-800.0d0)))) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = (4096.0d0 - (u ** 3.0d0)) / 256.0d0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+202) || !(l <= -800.0)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (4096.0 - Math.pow(U, 3.0)) / 256.0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.05e+202) or not (l <= -800.0):
		tmp = U + (J * (l * 2.0))
	else:
		tmp = (4096.0 - math.pow(U, 3.0)) / 256.0
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.05e+202) || !(l <= -800.0))
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(Float64(4096.0 - (U ^ 3.0)) / 256.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.05e+202) || ~((l <= -800.0)))
		tmp = U + (J * (l * 2.0));
	else
		tmp = (4096.0 - (U ^ 3.0)) / 256.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.05e+202], N[Not[LessEqual[l, -800.0]], $MachinePrecision]], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4096.0 - N[Power[U, 3.0], $MachinePrecision]), $MachinePrecision] / 256.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.05e202 or -800 < l

   1. Initial program 83.2%

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

   3. Taylor expanded in l around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*r* 73.4%

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

      3. associate-*l* 73.4%

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

      4. *-commutative 73.4%

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

      5. *-commutative 73.4%

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

      6. *-commutative 73.4%

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

      7. associate-*l* 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in K around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r* 63.6%

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

      3. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -1.05e202 < l < -800

   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 3.3%

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

      1. fmm-undef 3.3%

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

      2. metadata-eval 3.3%

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

   6. Simplified 3.3%

      \[\leadsto \color{blue}{16 - U} \]
   7. Step-by-step derivation

      1. flip3-- 5.5%

         \[\leadsto \color{blue}{\frac{{16}^{3} - {U}^{3}}{16 \cdot 16 + \left(U \cdot U + 16 \cdot U\right)}} \]

      2. metadata-eval 5.5%

         \[\leadsto \frac{\color{blue}{4096} - {U}^{3}}{16 \cdot 16 + \left(U \cdot U + 16 \cdot U\right)} \]

      3. metadata-eval 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{\color{blue}{256} + \left(U \cdot U + 16 \cdot U\right)} \]

      4. distribute-rgt-out 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{256 + \color{blue}{U \cdot \left(U + 16\right)}} \]

      5. +-commutative 5.5%

         \[\leadsto \frac{4096 - {U}^{3}}{256 + U \cdot \color{blue}{\left(16 + U\right)}} \]

   8. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{\frac{4096 - {U}^{3}}{256 + U \cdot \left(16 + U\right)}} \]

   9. Taylor expanded in U around 0 38.1%

      \[\leadsto \frac{4096 - {U}^{3}}{\color{blue}{256}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+202} \lor \neg \left(\ell \leq -800\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4096 - {U}^{3}}{256}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+198} \lor \neg \left(\ell \leq -2500\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{256 - U \cdot U}{U + 16}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.05e+198) (not (<= l -2500.0)))
   (+ U (* J (* l 2.0)))
   (/ (- 256.0 (* U U)) (+ U 16.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+198) || !(l <= -2500.0)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (256.0 - (U * U)) / (U + 16.0);
	}
	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 <= (-1.05d+198)) .or. (.not. (l <= (-2500.0d0)))) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = (256.0d0 - (u * u)) / (u + 16.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+198) || !(l <= -2500.0)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (256.0 - (U * U)) / (U + 16.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.05e+198) or not (l <= -2500.0):
		tmp = U + (J * (l * 2.0))
	else:
		tmp = (256.0 - (U * U)) / (U + 16.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.05e+198) || !(l <= -2500.0))
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(Float64(256.0 - Float64(U * U)) / Float64(U + 16.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.05e+198) || ~((l <= -2500.0)))
		tmp = U + (J * (l * 2.0));
	else
		tmp = (256.0 - (U * U)) / (U + 16.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.05e+198], N[Not[LessEqual[l, -2500.0]], $MachinePrecision]], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(256.0 - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + 16.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.05000000000000006e198 or -2500 < l

   1. Initial program 83.2%

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

   3. Taylor expanded in l around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*r* 73.4%

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

      3. associate-*l* 73.4%

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

      4. *-commutative 73.4%

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

      5. *-commutative 73.4%

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

      6. *-commutative 73.4%

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

      7. associate-*l* 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in K around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r* 63.6%

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

      3. *-commutative 63.6%

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

   8. Simplified 63.6%

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

   if -1.05000000000000006e198 < l < -2500

   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 3.3%

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

      1. fmm-undef 3.3%

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

      2. metadata-eval 3.3%

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

   6. Simplified 3.3%

      \[\leadsto \color{blue}{16 - U} \]
   7. Step-by-step derivation

      1. sub-neg 3.3%

         \[\leadsto \color{blue}{16 + \left(-U\right)} \]

      2. flip-+ 33.9%

         \[\leadsto \color{blue}{\frac{16 \cdot 16 - \left(-U\right) \cdot \left(-U\right)}{16 - \left(-U\right)}} \]

      3. metadata-eval 33.9%

         \[\leadsto \frac{\color{blue}{256} - \left(-U\right) \cdot \left(-U\right)}{16 - \left(-U\right)} \]

   8. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{\frac{256 - \left(-U\right) \cdot \left(-U\right)}{16 - \left(-U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+198} \lor \neg \left(\ell \leq -2500\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{256 - U \cdot U}{U + 16}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.0% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1000.0) (not (<= l 1.52e+29))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 1.52e+29)) {
		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 <= (-1000.0d0)) .or. (.not. (l <= 1.52d+29))) 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 <= -1000.0) || !(l <= 1.52e+29)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1000.0) or not (l <= 1.52e+29):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1000.0) || !(l <= 1.52e+29))
		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 <= -1000.0) || ~((l <= 1.52e+29)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1000.0], N[Not[LessEqual[l, 1.52e+29]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1e3 or 1.52e29 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 15.0%

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

   if -1e3 < l < 1.52e29

   1. Initial program 72.9%

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

   3. Taylor expanded in J around 0 69.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in l around 0 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 63.4%

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

   3. associate-*l* 63.4%

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

   4. *-commutative 63.4%

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

   5. *-commutative 63.4%

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

   6. *-commutative 63.4%

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

   7. associate-*l* 63.4%

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

5. Simplified 63.4%

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

6. Taylor expanded in K around 0 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*r* 54.9%

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

   3. *-commutative 54.9%

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

8. Simplified 54.9%

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

9. Final simplification 54.9%

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

Alternative 12: 36.3% 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 86.3%

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

3. Taylor expanded in J around 0 36.2%

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

Alternative 13: 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 86.3%

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

4. Applied egg-rr 22.6%

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

5. Taylor expanded in U around 0 2.9%

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

Reproduce

herbie shell --seed 2024182 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))