Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.3%

Time: 14.8s

Alternatives: 16

Speedup: 1.0×

• 50.4% 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: 86.7% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;J \cdot \left(t\_0 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-9)))
     (* J (* t_0 (cos (* K 0.5))))
     (+
      U
      (*
       (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* 2.0 J)))
       (cos (/ K 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-9)) {
		tmp = J * (t_0 * cos((K * 0.5)));
	} else {
		tmp = U + ((l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (2.0 * J))) * cos((K / 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e-9)) {
		tmp = J * (t_0 * Math.cos((K * 0.5)));
	} else {
		tmp = U + ((l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (2.0 * J))) * Math.cos((K / 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-9):
		tmp = J * (t_0 * math.cos((K * 0.5)))
	else:
		tmp = U + ((l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (2.0 * J))) * math.cos((K / 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-9))
		tmp = Float64(J * Float64(t_0 * cos(Float64(K * 0.5))));
	else
		tmp = Float64(U + Float64(Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(2.0 * J))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-9)))
		tmp = J * (t_0 * cos((K * 0.5)));
	else
		tmp = U + ((l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (2.0 * J))) * cos((K / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-9]], $MachinePrecision]], N[(J * N[(t$95$0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 5.0000000000000001e-9 < (-.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. Taylor expanded in J around 0 100.0%

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

   4. Taylor expanded in J around inf 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-9

   1. Initial program 73.0%

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

   3. Taylor expanded in l around 0 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.04:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.04)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
     (+ U (* 2.0 (* J (log1p (expm1 l))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.04) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} 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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.04) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.04:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	else:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.04)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.04], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $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))) < 0.0400000000000000008

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 89.9%

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

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0 59.0%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in K around 0 88.0%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\color{blue}{e^{\ell} - 1}\right)\right) + U \]
   7. Step-by-step derivation

      1. expm1-define 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (+ 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)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} 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)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} 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)) <= 0.04:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	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)) <= 0.04)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	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], 0.04], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $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))) < 0.0400000000000000008

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 66.5%

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

   4. Taylor expanded in U around inf 70.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 77.2%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   6. Simplified 77.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0 59.0%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in K around 0 88.0%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\color{blue}{e^{\ell} - 1}\right)\right) + U \]
   7. Step-by-step derivation

      1. expm1-define 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 60.6%

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

   1. log1p-expm1-u 99.7%

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

   2. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (* U (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / 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 (cos((k / 2.0d0)) <= 0.04d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    else
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    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)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.04:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	else:
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.04)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.04)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	else
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.04], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0400000000000000008

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 66.5%

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

   4. Taylor expanded in U around inf 70.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 77.2%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   6. Simplified 77.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0 88.2%

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

   4. Taylor expanded in K around 0 84.3%

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

   5. Taylor expanded in U around inf 85.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (+ 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 (cos((K / 2.0)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} 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 (cos((k / 2.0d0)) <= 0.04d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    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 (Math.cos((K / 2.0)) <= 0.04) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} 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 math.cos((K / 2.0)) <= 0.04:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	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 (cos(Float64(K / 2.0)) <= 0.04)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	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 (cos((K / 2.0)) <= 0.04)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	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[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.04], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $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 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0400000000000000008

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 66.5%

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

   4. Taylor expanded in U around inf 70.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 77.2%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   6. Simplified 77.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0 88.2%

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

   4. Taylor expanded in K around 0 84.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (+ 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 (cos((K / 2.0)) <= 0.04) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} 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 (cos((k / 2.0d0)) <= 0.04d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    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 (Math.cos((K / 2.0)) <= 0.04) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} 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 math.cos((K / 2.0)) <= 0.04:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	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 (cos(Float64(K / 2.0)) <= 0.04)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	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 (cos((K / 2.0)) <= 0.04)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	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[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.04], 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[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0400000000000000008

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 66.5%

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

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in l around 0 88.2%

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

   4. Taylor expanded in K around 0 84.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{elif}\;\ell \leq -920000000 \lor \neg \left(\ell \leq 2.9 \cdot 10^{+30}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -3.9e+129)
     t_0
     (if (<= l -1.45e+112)
       (+ U (* J (+ -0.5 (* (pow K 2.0) 0.0625))))
       (if (or (<= l -920000000.0) (not (<= l 2.9e+30)))
         t_0
         (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -1.45e+112) {
		tmp = U + (J * (-0.5 + (pow(K, 2.0) * 0.0625)));
	} else if ((l <= -920000000.0) || !(l <= 2.9e+30)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	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 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-3.9d+129)) then
        tmp = t_0
    else if (l <= (-1.45d+112)) then
        tmp = u + (j * ((-0.5d0) + ((k ** 2.0d0) * 0.0625d0)))
    else if ((l <= (-920000000.0d0)) .or. (.not. (l <= 2.9d+30))) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -1.45e+112) {
		tmp = U + (J * (-0.5 + (Math.pow(K, 2.0) * 0.0625)));
	} else if ((l <= -920000000.0) || !(l <= 2.9e+30)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -3.9e+129:
		tmp = t_0
	elif l <= -1.45e+112:
		tmp = U + (J * (-0.5 + (math.pow(K, 2.0) * 0.0625)))
	elif (l <= -920000000.0) or not (l <= 2.9e+30):
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -3.9e+129)
		tmp = t_0;
	elseif (l <= -1.45e+112)
		tmp = Float64(U + Float64(J * Float64(-0.5 + Float64((K ^ 2.0) * 0.0625))));
	elseif ((l <= -920000000.0) || !(l <= 2.9e+30))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -3.9e+129)
		tmp = t_0;
	elseif (l <= -1.45e+112)
		tmp = U + (J * (-0.5 + ((K ^ 2.0) * 0.0625)));
	elseif ((l <= -920000000.0) || ~((l <= 2.9e+30)))
		tmp = t_0;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.9e+129], t$95$0, If[LessEqual[l, -1.45e+112], N[(U + N[(J * N[(-0.5 + N[(N[Power[K, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -920000000.0], N[Not[LessEqual[l, 2.9e+30]], $MachinePrecision]], t$95$0, N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8999999999999997e129 or -1.4500000000000001e112 < l < -9.2e8 or 2.8999999999999998e30 < 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 81.4%

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

   4. Taylor expanded in K around 0 69.5%

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

   5. Taylor expanded in l around inf 69.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r* 69.6%

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

   7. Simplified 69.6%

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

   if -3.8999999999999997e129 < l < -1.4500000000000001e112

   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 3.8%

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

   5. Taylor expanded in K around 0 80.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 80.5%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 80.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 80.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 80.5%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 80.5%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   if -9.2e8 < l < 2.8999999999999998e30

   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 l around 0 95.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{elif}\;\ell \leq -920000000 \lor \neg \left(\ell \leq 2.9 \cdot 10^{+30}\right):\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{+42} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+92}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -3.9e+129)
     t_0
     (if (<= l -1.45e+112)
       (+ U (* J (+ -0.5 (* (pow K 2.0) 0.0625))))
       (if (or (<= l -8.6e+42) (not (<= l 1.9e+92)))
         t_0
         (* U (+ 1.0 (* 2.0 (* J (/ l U))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -1.45e+112) {
		tmp = U + (J * (-0.5 + (pow(K, 2.0) * 0.0625)));
	} else if ((l <= -8.6e+42) || !(l <= 1.9e+92)) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / 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) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-3.9d+129)) then
        tmp = t_0
    else if (l <= (-1.45d+112)) then
        tmp = u + (j * ((-0.5d0) + ((k ** 2.0d0) * 0.0625d0)))
    else if ((l <= (-8.6d+42)) .or. (.not. (l <= 1.9d+92))) then
        tmp = t_0
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -1.45e+112) {
		tmp = U + (J * (-0.5 + (Math.pow(K, 2.0) * 0.0625)));
	} else if ((l <= -8.6e+42) || !(l <= 1.9e+92)) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -3.9e+129:
		tmp = t_0
	elif l <= -1.45e+112:
		tmp = U + (J * (-0.5 + (math.pow(K, 2.0) * 0.0625)))
	elif (l <= -8.6e+42) or not (l <= 1.9e+92):
		tmp = t_0
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -3.9e+129)
		tmp = t_0;
	elseif (l <= -1.45e+112)
		tmp = Float64(U + Float64(J * Float64(-0.5 + Float64((K ^ 2.0) * 0.0625))));
	elseif ((l <= -8.6e+42) || !(l <= 1.9e+92))
		tmp = t_0;
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -3.9e+129)
		tmp = t_0;
	elseif (l <= -1.45e+112)
		tmp = U + (J * (-0.5 + ((K ^ 2.0) * 0.0625)));
	elseif ((l <= -8.6e+42) || ~((l <= 1.9e+92)))
		tmp = t_0;
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.9e+129], t$95$0, If[LessEqual[l, -1.45e+112], N[(U + N[(J * N[(-0.5 + N[(N[Power[K, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -8.6e+42], N[Not[LessEqual[l, 1.9e+92]], $MachinePrecision]], t$95$0, N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8999999999999997e129 or -1.4500000000000001e112 < l < -8.5999999999999996e42 or 1.9e92 < 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 90.2%

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

   4. Taylor expanded in K around 0 75.9%

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

   5. Taylor expanded in l around inf 75.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 75.9%

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

   7. Simplified 75.9%

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

   if -3.8999999999999997e129 < l < -1.4500000000000001e112

   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 3.8%

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

   5. Taylor expanded in K around 0 80.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 80.5%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 80.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 80.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 80.5%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 80.5%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   if -8.5999999999999996e42 < l < 1.9e92

   1. Initial program 78.0%

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

   3. Taylor expanded in l around 0 82.3%

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

   4. Taylor expanded in K around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.4%

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

      4. *-commutative 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in U around inf 75.0%

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

      1. associate-/l* 76.2%

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

   9. Simplified 76.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{+42} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+92}\right):\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+43} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+92}\right):\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.5e+43) (not (<= l 1.7e+92)))
   (* (pow l 3.0) (* J 0.3333333333333333))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.5e+43) || !(l <= 1.7e+92)) {
		tmp = pow(l, 3.0) * (J * 0.3333333333333333);
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / 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 <= (-1.5d+43)) .or. (.not. (l <= 1.7d+92))) then
        tmp = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.5e+43) || !(l <= 1.7e+92)) {
		tmp = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.5e+43) or not (l <= 1.7e+92):
		tmp = math.pow(l, 3.0) * (J * 0.3333333333333333)
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.5e+43) || !(l <= 1.7e+92))
		tmp = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.5e+43) || ~((l <= 1.7e+92)))
		tmp = (l ^ 3.0) * (J * 0.3333333333333333);
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.5e+43], N[Not[LessEqual[l, 1.7e+92]], $MachinePrecision]], N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.50000000000000008e43 or 1.6999999999999999e92 < 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 90.6%

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

   4. Taylor expanded in K around 0 72.5%

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

   5. Taylor expanded in l around inf 72.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r* 72.5%

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

   7. Simplified 72.5%

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

   if -1.50000000000000008e43 < l < 1.6999999999999999e92

   1. Initial program 78.0%

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

   3. Taylor expanded in l around 0 82.3%

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

   4. Taylor expanded in K around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.4%

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

      4. *-commutative 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in U around inf 75.0%

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

      1. associate-/l* 76.2%

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

   9. Simplified 76.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+43} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+92}\right):\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.6% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 6.4 \cdot 10^{-28}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1000.0) (not (<= l 6.4e-28))) (* l (* 2.0 J)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 6.4e-28)) {
		tmp = l * (2.0 * J);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1000.0d0)) .or. (.not. (l <= 6.4d-28))) then
        tmp = l * (2.0d0 * j)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 6.4e-28)) {
		tmp = l * (2.0 * J);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1000.0) or not (l <= 6.4e-28):
		tmp = l * (2.0 * J)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1000.0) || !(l <= 6.4e-28))
		tmp = Float64(l * Float64(2.0 * J));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1000.0) || ~((l <= 6.4e-28)))
		tmp = l * (2.0 * J);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1000.0], N[Not[LessEqual[l, 6.4e-28]], $MachinePrecision]], N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1e3 or 6.39999999999999964e-28 < l

   1. Initial program 98.2%

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

   3. Taylor expanded in l around 0 29.5%

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

   4. Taylor expanded in K around 0 24.2%

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

      1. +-commutative 24.2%

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

      2. *-commutative 24.2%

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

      3. associate-*r* 24.2%

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

      4. *-commutative 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in J around inf 23.9%

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

      1. associate-*r* 23.9%

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

      2. *-commutative 23.9%

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

   9. Simplified 23.9%

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

   if -1e3 < l < 6.39999999999999964e-28

   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 J around 0 73.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 6.4 \cdot 10^{-28}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.6% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+43} \lor \neg \left(\ell \leq 1.7 \cdot 10^{-9}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.4e+43) (not (<= l 1.7e-9))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.4e+43) || !(l <= 1.7e-9)) {
		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 <= (-2.4d+43)) .or. (.not. (l <= 1.7d-9))) 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 <= -2.4e+43) || !(l <= 1.7e-9)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.4e+43) or not (l <= 1.7e-9):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.4e+43) || !(l <= 1.7e-9))
		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 <= -2.4e+43) || ~((l <= 1.7e-9)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.4e+43], N[Not[LessEqual[l, 1.7e-9]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.40000000000000023e43 or 1.6999999999999999e-9 < l

   1. Initial program 99.3%

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

   4. Applied egg-rr 13.7%

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

   if -2.40000000000000023e43 < l < 1.6999999999999999e-9

   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 J around 0 67.1%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+43} \lor \neg \left(\ell \leq 1.7 \cdot 10^{-9}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l / 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 * (1.0d0 + (2.0d0 * (j * (l / u))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l / U))));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * (J * (l / U))))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * (J * (l / U))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 60.6%

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

4. Taylor expanded in K around 0 51.4%

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

   1. +-commutative 51.4%

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

   2. *-commutative 51.4%

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

   3. associate-*r* 51.4%

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

   4. *-commutative 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in U around inf 58.9%

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

   1. associate-/l* 61.5%

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

9. Simplified 61.5%

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

10. Final simplification 61.5%

    \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \]
11. Add Preprocessing

Alternative 14: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 * (2.0d0 * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 60.6%

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

4. Taylor expanded in K around 0 51.4%

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

   1. +-commutative 51.4%

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

   2. *-commutative 51.4%

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

   3. associate-*r* 51.4%

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

   4. *-commutative 51.4%

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

6. Simplified 51.4%

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

7. Final simplification 51.4%

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

Alternative 15: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

4. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

6. Simplified 2.8%

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

7. Final simplification 2.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 16: 36.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in J around 0 33.8%

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

4. Final simplification 33.8%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))