Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.5%

Time: 16.8s

Alternatives: 13

Speedup: 1.4×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* 2.0 (* J (log1p (expm1 (* l (cos (* 0.5 K))))))) U))

C

double code(double J, double l, double K, double U) {
	return (2.0 * (J * log1p(expm1((l * cos((0.5 * K))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (2.0 * (J * Math.log1p(Math.expm1((l * Math.cos((0.5 * K))))))) + U;
}

Python

def code(J, l, K, U):
	return (2.0 * (J * math.log1p(math.expm1((l * math.cos((0.5 * K))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(2.0 * Float64(J * log1p(expm1(Float64(l * cos(Float64(0.5 * K))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Log[1 + N[(Exp[N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $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 60.1%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

5. Applied egg-rr 99.3%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]
6. Add Preprocessing

Alternative 2: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
   (+ U (* 2.0 (* J (log1p (expm1 l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else {
		tmp = U + (2.0 * (J * log1p(expm1(l))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.01:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	else:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0 66.4%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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 58.1%

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

      1. log1p-expm1-u 99.1%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   5. Applied egg-rr 99.1%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 95.0%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \mathsf{log1p}\left(\mathsf{expm1}\left(J \cdot \left(2 \cdot \ell\right)\right)\right)\\ t_1 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -850000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 290:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (log1p (expm1 (* J (* 2.0 l))))))
        (t_1 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -3.5e+135)
     t_1
     (if (<= l -850000.0)
       t_0
       (if (<= l 290.0)
         (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
         (if (<= l 8.8e+104) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + log1p(expm1((J * (2.0 * l))));
	double t_1 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -3.5e+135) {
		tmp = t_1;
	} else if (l <= -850000.0) {
		tmp = t_0;
	} else if (l <= 290.0) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else if (l <= 8.8e+104) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + Math.log1p(Math.expm1((J * (2.0 * l))));
	double t_1 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -3.5e+135) {
		tmp = t_1;
	} else if (l <= -850000.0) {
		tmp = t_0;
	} else if (l <= 290.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else if (l <= 8.8e+104) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + math.log1p(math.expm1((J * (2.0 * l))))
	t_1 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -3.5e+135:
		tmp = t_1
	elif l <= -850000.0:
		tmp = t_0
	elif l <= 290.0:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	elif l <= 8.8e+104:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + log1p(expm1(Float64(J * Float64(2.0 * l)))))
	t_1 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -3.5e+135)
		tmp = t_1;
	elseif (l <= -850000.0)
		tmp = t_0;
	elseif (l <= 290.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	elseif (l <= 8.8e+104)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[Log[1 + N[(Exp[N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.5e+135], t$95$1, If[LessEqual[l, -850000.0], t$95$0, If[LessEqual[l, 290.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.8e+104], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5000000000000003e135 or 8.80000000000000002e104 < 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 0 100.0%

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

   4. Taylor expanded in K around 0 71.6%

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

   5. Taylor expanded in l around inf 71.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]

   if -3.5000000000000003e135 < l < -8.5e5 or 290 < l < 8.80000000000000002e104

   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 78.6%

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

   4. Taylor expanded in l around 0 8.5%

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

      1. associate-*r* 8.5%

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

   6. Simplified 8.5%

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

      1. log1p-expm1-u 47.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot J\right) \cdot \ell\right)\right)} + U \]

      2. *-commutative 47.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell\right)\right) + U \]

      3. associate-*l* 47.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}\right)\right) + U \]

      4. *-commutative 47.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(J \cdot \color{blue}{\left(\ell \cdot 2\right)}\right)\right) + U \]

   8. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(J \cdot \left(\ell \cdot 2\right)\right)\right)} + U \]

   if -8.5e5 < l < 290

   1. Initial program 73.1%

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

   3. Taylor expanded in l around 0 98.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+135}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -850000:\\ \;\;\;\;U + \mathsf{log1p}\left(\mathsf{expm1}\left(J \cdot \left(2 \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 290:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+104}:\\ \;\;\;\;U + \mathsf{log1p}\left(\mathsf{expm1}\left(J \cdot \left(2 \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-28}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-28)
   (+ U (* 2.0 (* J (log1p (expm1 l)))))
   (+
    U
    (*
     (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-28) {
		tmp = U + (2.0 * (J * log1p(expm1(l))));
	} else {
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) * cos((K / 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-28) {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	} else {
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) * Math.cos((K / 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 1e-28:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	else:
		tmp = U + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) * math.cos((K / 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-28)
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-28], N[(U + N[(2.0 * N[(J * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 9.99999999999999971e-29

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0 58.3%

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

      1. log1p-expm1-u 99.0%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   5. Applied egg-rr 99.0%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 83.9%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right)\right) + U \]

   if 9.99999999999999971e-29 < (/.f64 K #s(literal 2 binary64))

   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 91.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-28}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0 65.7%

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

      1. associate-*r* 35.4%

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

   5. Simplified 65.7%

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

   6. Taylor expanded in K around 0 41.7%

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

   7. Taylor expanded in K around inf 57.3%

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

      1. associate-*r* 57.3%

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

      2. *-commutative 57.3%

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

      3. *-commutative 57.3%

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

   9. Simplified 57.3%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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 84.0%

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

   4. Taylor expanded in K around 0 78.0%

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

   5. Taylor expanded in l around inf 69.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-159}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.1e+56)
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (if (<= l 3.7e-159)
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
     (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.1e+56) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else if (l <= 3.7e-159) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-3.1d+56)) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else if (l <= 3.7d-159) then
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.1e+56) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else if (l <= 3.7e-159) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.1e+56:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	elif l <= 3.7e-159:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.1e+56)
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	elseif (l <= 3.7e-159)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.1e+56)
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	elseif (l <= 3.7e-159)
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.1e+56], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e-159], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.10000000000000005e56

   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 87.9%

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

   4. Taylor expanded in K around 0 62.9%

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

   5. Taylor expanded in l around inf 62.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]

   if -3.10000000000000005e56 < l < 3.6999999999999999e-159

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 89.8%

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

   if 3.6999999999999999e-159 < l

   1. Initial program 93.2%

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

   3. Taylor expanded in l around 0 79.0%

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

   4. Taylor expanded in K around 0 62.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-159}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+117} \lor \neg \left(\ell \leq 0.00182\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7.2e+117) (not (<= l 0.00182)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.2e+117) || !(l <= 0.00182)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * 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 <= (-7.2d+117)) .or. (.not. (l <= 0.00182d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.2e+117) || !(l <= 0.00182)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7.2e+117) or not (l <= 0.00182):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7.2e+117) || !(l <= 0.00182))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7.2e+117) || ~((l <= 0.00182)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7.2e+117], N[Not[LessEqual[l, 0.00182]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.20000000000000025e117 or 0.00182 < 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 80.9%

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

   4. Taylor expanded in K around 0 58.0%

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

   5. Taylor expanded in l around inf 58.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]

   if -7.20000000000000025e117 < l < 0.00182

   1. Initial program 77.4%

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

   3. Taylor expanded in l around 0 84.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+117} \lor \neg \left(\ell \leq 0.00182\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+34}:\\ \;\;\;\;U - J \cdot {K}^{2}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{8}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(8 - {K}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.3e+34)
   (- U (* J (pow K 2.0)))
   (if (<= l 2.05e-23)
     (+ U (* l (* 2.0 J)))
     (if (<= l 1.5e+137)
       (* U (+ 1.0 (* J (/ 8.0 U))))
       (* J (- 8.0 (pow K 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.3e+34) {
		tmp = U - (J * pow(K, 2.0));
	} else if (l <= 2.05e-23) {
		tmp = U + (l * (2.0 * J));
	} else if (l <= 1.5e+137) {
		tmp = U * (1.0 + (J * (8.0 / U)));
	} else {
		tmp = J * (8.0 - pow(K, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-2.3d+34)) then
        tmp = u - (j * (k ** 2.0d0))
    else if (l <= 2.05d-23) then
        tmp = u + (l * (2.0d0 * j))
    else if (l <= 1.5d+137) then
        tmp = u * (1.0d0 + (j * (8.0d0 / u)))
    else
        tmp = j * (8.0d0 - (k ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.3e+34) {
		tmp = U - (J * Math.pow(K, 2.0));
	} else if (l <= 2.05e-23) {
		tmp = U + (l * (2.0 * J));
	} else if (l <= 1.5e+137) {
		tmp = U * (1.0 + (J * (8.0 / U)));
	} else {
		tmp = J * (8.0 - Math.pow(K, 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.3e+34:
		tmp = U - (J * math.pow(K, 2.0))
	elif l <= 2.05e-23:
		tmp = U + (l * (2.0 * J))
	elif l <= 1.5e+137:
		tmp = U * (1.0 + (J * (8.0 / U)))
	else:
		tmp = J * (8.0 - math.pow(K, 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.3e+34)
		tmp = Float64(U - Float64(J * (K ^ 2.0)));
	elseif (l <= 2.05e-23)
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	elseif (l <= 1.5e+137)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(8.0 / U))));
	else
		tmp = Float64(J * Float64(8.0 - (K ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.3e+34)
		tmp = U - (J * (K ^ 2.0));
	elseif (l <= 2.05e-23)
		tmp = U + (l * (2.0 * J));
	elseif (l <= 1.5e+137)
		tmp = U * (1.0 + (J * (8.0 / U)));
	else
		tmp = J * (8.0 - (K ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.3e+34], N[(U - N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e-23], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+137], N[(U * N[(1.0 + N[(J * N[(8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(8.0 - N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.2999999999999998e34

   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 1.5%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 26.5%

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

      1. +-commutative 26.5%

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

      2. mul-1-neg 26.5%

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

      3. unsub-neg 26.5%

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

      4. *-commutative 26.5%

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

   7. Simplified 26.5%

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

   8. Taylor expanded in K around inf 26.9%

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

      1. mul-1-neg 26.9%

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

      2. distribute-rgt-neg-out 26.9%

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

   10. Simplified 26.9%

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

   if -2.2999999999999998e34 < l < 2.05000000000000015e-23

   1. Initial program 74.1%

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

   3. Taylor expanded in K around 0 72.2%

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

   4. Taylor expanded in l around 0 81.7%

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

      1. associate-*r* 81.0%

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

   6. Simplified 81.0%

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

   if 2.05000000000000015e-23 < l < 1.5e137

   1. Initial program 98.4%

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

   4. Applied egg-rr 12.2%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in U around inf 27.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + 8 \cdot \frac{J \cdot \cos \left(0.5 \cdot K\right)}{U}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 27.7%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{8 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)}{U}}\right) \]

      2. *-commutative 27.7%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 8}}{U}\right) \]

      3. associate-*r* 27.7%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 8\right)}}{U}\right) \]

   7. Simplified 27.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 8\right)}{U}\right)} \]

   8. Taylor expanded in K around 0 24.9%

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

      1. associate-*r/ 24.9%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{8 \cdot J}{U}}\right) \]

      2. *-commutative 24.9%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot 8}}{U}\right) \]

      3. associate-*r/ 24.9%

         \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{8}{U}}\right) \]

   10. Simplified 24.9%

       \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{8}{U}}\right) \]

   if 1.5e137 < 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 3.3%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around inf 17.2%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in J around inf 3.1%

      \[\leadsto \color{blue}{8 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 3.1%

         \[\leadsto \color{blue}{\left(8 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      2. *-commutative 3.1%

         \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(8 \cdot J\right)} \]

      3. *-commutative 3.1%

         \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot 8\right)} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 8\right)} \]

   9. Taylor expanded in K around 0 27.5%

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

       1. associate-*r* 27.5%

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

       2. +-commutative 27.5%

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

       3. mul-1-neg 27.5%

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

       4. cancel-sign-sub-inv 27.5%

          \[\leadsto \color{blue}{8 \cdot J - J \cdot {K}^{2}} \]

       5. *-commutative 27.5%

          \[\leadsto 8 \cdot J - \color{blue}{{K}^{2} \cdot J} \]

       6. distribute-rgt-out-- 27.5%

          \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} \]

   11. Simplified 27.5%

       \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+34}:\\ \;\;\;\;U - J \cdot {K}^{2}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{8}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(8 - {K}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(8 - {K}^{2}\right)\\ \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+137}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{8}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- 8.0 (pow K 2.0)))))
   (if (<= l -2.85e+20)
     t_0
     (if (<= l 2.05e-23)
       (+ U (* l (* 2.0 J)))
       (if (<= l 7.5e+137) (* U (+ 1.0 (* J (/ 8.0 U)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (8.0 - pow(K, 2.0));
	double tmp;
	if (l <= -2.85e+20) {
		tmp = t_0;
	} else if (l <= 2.05e-23) {
		tmp = U + (l * (2.0 * J));
	} else if (l <= 7.5e+137) {
		tmp = U * (1.0 + (J * (8.0 / U)));
	} 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 = j * (8.0d0 - (k ** 2.0d0))
    if (l <= (-2.85d+20)) then
        tmp = t_0
    else if (l <= 2.05d-23) then
        tmp = u + (l * (2.0d0 * j))
    else if (l <= 7.5d+137) then
        tmp = u * (1.0d0 + (j * (8.0d0 / u)))
    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 = J * (8.0 - Math.pow(K, 2.0));
	double tmp;
	if (l <= -2.85e+20) {
		tmp = t_0;
	} else if (l <= 2.05e-23) {
		tmp = U + (l * (2.0 * J));
	} else if (l <= 7.5e+137) {
		tmp = U * (1.0 + (J * (8.0 / U)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (8.0 - math.pow(K, 2.0))
	tmp = 0
	if l <= -2.85e+20:
		tmp = t_0
	elif l <= 2.05e-23:
		tmp = U + (l * (2.0 * J))
	elif l <= 7.5e+137:
		tmp = U * (1.0 + (J * (8.0 / U)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(8.0 - (K ^ 2.0)))
	tmp = 0.0
	if (l <= -2.85e+20)
		tmp = t_0;
	elseif (l <= 2.05e-23)
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	elseif (l <= 7.5e+137)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(8.0 / U))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (8.0 - (K ^ 2.0));
	tmp = 0.0;
	if (l <= -2.85e+20)
		tmp = t_0;
	elseif (l <= 2.05e-23)
		tmp = U + (l * (2.0 * J));
	elseif (l <= 7.5e+137)
		tmp = U * (1.0 + (J * (8.0 / U)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(8.0 - N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.85e+20], t$95$0, If[LessEqual[l, 2.05e-23], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+137], N[(U * N[(1.0 + N[(J * N[(8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.85e20 or 7.50000000000000025e137 < 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 2.0%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around inf 10.4%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in J around inf 1.4%

      \[\leadsto \color{blue}{8 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 1.4%

         \[\leadsto \color{blue}{\left(8 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      2. *-commutative 1.4%

         \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(8 \cdot J\right)} \]

      3. *-commutative 1.4%

         \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot 8\right)} \]

   8. Simplified 1.4%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 8\right)} \]

   9. Taylor expanded in K around 0 26.8%

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

       1. associate-*r* 26.8%

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

       2. +-commutative 26.8%

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

       3. mul-1-neg 26.8%

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

       4. cancel-sign-sub-inv 26.8%

          \[\leadsto \color{blue}{8 \cdot J - J \cdot {K}^{2}} \]

       5. *-commutative 26.8%

          \[\leadsto 8 \cdot J - \color{blue}{{K}^{2} \cdot J} \]

       6. distribute-rgt-out-- 26.8%

          \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} \]

   11. Simplified 26.8%

       \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} \]

   if -2.85e20 < l < 2.05000000000000015e-23

   1. Initial program 73.4%

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

   3. Taylor expanded in K around 0 72.3%

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

   4. Taylor expanded in l around 0 83.6%

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

      1. associate-*r* 82.8%

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

   6. Simplified 82.8%

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

   if 2.05000000000000015e-23 < l < 7.50000000000000025e137

   1. Initial program 98.4%

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

   4. Applied egg-rr 12.2%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in U around inf 27.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + 8 \cdot \frac{J \cdot \cos \left(0.5 \cdot K\right)}{U}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 27.7%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{8 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)}{U}}\right) \]

      2. *-commutative 27.7%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 8}}{U}\right) \]

      3. associate-*r* 27.7%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 8\right)}}{U}\right) \]

   7. Simplified 27.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 8\right)}{U}\right)} \]

   8. Taylor expanded in K around 0 24.9%

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

      1. associate-*r/ 24.9%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{8 \cdot J}{U}}\right) \]

      2. *-commutative 24.9%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot 8}}{U}\right) \]

      3. associate-*r/ 24.9%

         \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{8}{U}}\right) \]

   10. Simplified 24.9%

       \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{8}{U}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+20}:\\ \;\;\;\;J \cdot \left(8 - {K}^{2}\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+137}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{8}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(8 - {K}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+56} \lor \neg \left(\ell \leq 2 \cdot 10^{-76}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.1e+56) (not (<= l 2e-76)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* l (* 2.0 J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e+56) || !(l <= 2e-76)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (l * (2.0 * 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+56)) .or. (.not. (l <= 2d-76))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (l * (2.0d0 * 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+56) || !(l <= 2e-76)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (l * (2.0 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.1e+56) or not (l <= 2e-76):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (l * (2.0 * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.1e+56) || !(l <= 2e-76))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.1e+56) || ~((l <= 2e-76)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (l * (2.0 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.1e+56], N[Not[LessEqual[l, 2e-76]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.10000000000000017e56 or 1.99999999999999985e-76 < l

   1. Initial program 97.4%

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

   3. Taylor expanded in l around 0 80.5%

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

   4. Taylor expanded in K around 0 60.1%

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

   5. Taylor expanded in l around inf 57.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]

   if -2.10000000000000017e56 < l < 1.99999999999999985e-76

   1. Initial program 74.3%

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

   3. Taylor expanded in K around 0 72.3%

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

   4. Taylor expanded in l around 0 77.7%

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

      1. associate-*r* 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+56} \lor \neg \left(\ell \leq 2 \cdot 10^{-76}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.2% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(8 + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 7.5e-20) U (* J (+ 8.0 (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 7.5e-20) {
		tmp = U;
	} else {
		tmp = J * (8.0 + (U / 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 <= 7.5d-20) then
        tmp = u
    else
        tmp = j * (8.0d0 + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 7.5e-20) {
		tmp = U;
	} else {
		tmp = J * (8.0 + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 7.5e-20:
		tmp = U
	else:
		tmp = J * (8.0 + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 7.5e-20)
		tmp = U;
	else
		tmp = Float64(J * Float64(8.0 + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 7.5e-20)
		tmp = U;
	else
		tmp = J * (8.0 + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 7.5e-20], U, N[(J * N[(8.0 + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.49999999999999981e-20

   1. Initial program 82.3%

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

   4. Applied egg-rr 36.7%

      \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around 0 46.8%

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

   if 7.49999999999999981e-20 < l

   1. Initial program 99.0%

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

   4. Applied egg-rr 7.3%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around inf 14.3%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in K around 0 13.6%

      \[\leadsto \color{blue}{J \cdot \left(8 + \frac{U}{J}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(l * N[(2.0 * 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 K around 0 73.6%

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

4. Taylor expanded in l around 0 48.9%

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

   1. associate-*r* 48.6%

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

6. Simplified 48.6%

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

7. Final simplification 48.6%

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

Alternative 13: 37.6% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 28.5%

   \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Taylor expanded in J around 0 36.1%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))