Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.3%

Time: 16.6s

Alternatives: 19

Speedup: 1.3×

• 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(J, t\_0 \cdot t\_1, U\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot t\_1, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 -1.0)
     (fma J (* t_0 t_1) U)
     (if (<= t_0 5e-9)
       (fma J (* (* 2.0 l) t_1) U)
       (* J (* (cos (* 0.5 K)) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = fma(J, (t_0 * t_1), U);
	} else if (t_0 <= 5e-9) {
		tmp = fma(J, ((2.0 * l) * t_1), U);
	} else {
		tmp = J * (cos((0.5 * K)) * t_0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = fma(J, Float64(t_0 * t_1), U);
	elseif (t_0 <= 5e-9)
		tmp = fma(J, Float64(Float64(2.0 * l) * t_1), U);
	else
		tmp = Float64(J * Float64(cos(Float64(0.5 * K)) * t_0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(J * N[(t$95$0 * t$95$1), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 5e-9], N[(J * N[(N[(2.0 * l), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if -1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-9

   1. Initial program 72.5%

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

      1. associate-*l* 72.5%

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

      2. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in l around 0 99.9%

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

   if 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 inf 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* J (* (cos (* 0.5 K)) t_0))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 5e-9)
       (+
        (*
         (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* 2.0 J)))
         (cos (/ K 2.0)))
        U)
       t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = J * (cos((0.5 * K)) * t_0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 5e-9) {
		tmp = ((l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (2.0 * J))) * cos((K / 2.0))) + U;
	} else {
		tmp = t_1;
	}
	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 t_1 = J * (Math.cos((0.5 * K)) * t_0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 5e-9) {
		tmp = ((l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (2.0 * J))) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = J * (math.cos((0.5 * K)) * t_0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 5e-9:
		tmp = ((l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (2.0 * J))) * math.cos((K / 2.0))) + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(J * Float64(cos(Float64(0.5 * K)) * t_0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 5e-9)
		tmp = Float64(Float64(Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(2.0 * J))) * cos(Float64(K / 2.0))) + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = J * (cos((0.5 * K)) * t_0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 5e-9)
		tmp = ((l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (2.0 * J))) * cos((K / 2.0))) + U;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(J * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 5e-9], N[(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] + U), $MachinePrecision], t$95$1]]]]

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 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. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\left(J \cdot t\_1\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot t\_0, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 -1.0)
     (+ (* (* J t_1) t_0) U)
     (if (<= t_1 5e-9)
       (fma J (* (* 2.0 l) t_0) U)
       (* J (* (cos (* 0.5 K)) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = ((J * t_1) * t_0) + U;
	} else if (t_1 <= 5e-9) {
		tmp = fma(J, ((2.0 * l) * t_0), U);
	} else {
		tmp = J * (cos((0.5 * K)) * t_1);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(Float64(J * t_1) * t_0) + U);
	elseif (t_1 <= 5e-9)
		tmp = fma(J, Float64(Float64(2.0 * l) * t_0), U);
	else
		tmp = Float64(J * Float64(cos(Float64(0.5 * K)) * t_1));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[(J * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$1, 5e-9], N[(J * N[(N[(2.0 * l), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1

   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

   if -1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-9

   1. Initial program 72.5%

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

      1. associate-*l* 72.5%

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

      2. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in l around 0 99.9%

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

   if 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 inf 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := t\_0 + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (+ t_0 U)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 1e+35) (fma J (* (* 2.0 l) (cos (/ K 2.0))) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = t_0 + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 1e+35) {
		tmp = fma(J, ((2.0 * l) * cos((K / 2.0))), U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = Float64(t_0 + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 1e+35)
		tmp = fma(J, Float64(Float64(2.0 * l) * cos(Float64(K / 2.0))), U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 1e+35], N[(J * N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0 or 9.9999999999999997e34 < (*.f64 J (-.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 K around 0 81.9%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 9.9999999999999997e34

   1. Initial program 73.0%

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

      1. associate-*l* 73.0%

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

      2. fma-define 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in l around 0 99.3%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := J \cdot t\_0 + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (+ (* J t_0) U)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 5e-9) (+ (* 2.0 (* J (* l (cos (* 0.5 K))))) U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = (J * t_0) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 5e-9) {
		tmp = (2.0 * (J * (l * cos((0.5 * K))))) + U;
	} else {
		tmp = t_1;
	}
	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 t_1 = (J * t_0) + U;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 5e-9) {
		tmp = (2.0 * (J * (l * Math.cos((0.5 * K))))) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = (J * t_0) + U
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 5e-9:
		tmp = (2.0 * (J * (l * math.cos((0.5 * K))))) + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(Float64(J * t_0) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 5e-9)
		tmp = Float64(Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))) + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = (J * t_0) + U;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 5e-9)
		tmp = (2.0 * (J * (l * cos((0.5 * K))))) + U;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(J * t$95$0), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 5e-9], N[(N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]

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 K around 0 81.9%

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

   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%

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

Alternative 6: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_0 + U\\ t_2 := J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{+25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.03:\\ \;\;\;\;\left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ (* (* J (* 0.3333333333333333 (pow l 3.0))) t_0) U))
        (t_2 (+ (* J (- (exp l) (exp (- l)))) U)))
   (if (<= l -5.6e+111)
     t_1
     (if (<= l -1.46e+25)
       t_2
       (if (<= l 0.03)
         (+
          (* (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* 2.0 J))) t_0)
          U)
         (if (<= l 6e+92) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((J * (0.3333333333333333 * pow(l, 3.0))) * t_0) + U;
	double t_2 = (J * (exp(l) - exp(-l))) + U;
	double tmp;
	if (l <= -5.6e+111) {
		tmp = t_1;
	} else if (l <= -1.46e+25) {
		tmp = t_2;
	} else if (l <= 0.03) {
		tmp = ((l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (2.0 * J))) * t_0) + U;
	} else if (l <= 6e+92) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = ((j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_0) + u
    t_2 = (j * (exp(l) - exp(-l))) + u
    if (l <= (-5.6d+111)) then
        tmp = t_1
    else if (l <= (-1.46d+25)) then
        tmp = t_2
    else if (l <= 0.03d0) then
        tmp = ((l * ((0.3333333333333333d0 * (j * (l ** 2.0d0))) + (2.0d0 * j))) * t_0) + u
    else if (l <= 6d+92) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_0) + U;
	double t_2 = (J * (Math.exp(l) - Math.exp(-l))) + U;
	double tmp;
	if (l <= -5.6e+111) {
		tmp = t_1;
	} else if (l <= -1.46e+25) {
		tmp = t_2;
	} else if (l <= 0.03) {
		tmp = ((l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (2.0 * J))) * t_0) + U;
	} else if (l <= 6e+92) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((J * (0.3333333333333333 * math.pow(l, 3.0))) * t_0) + U
	t_2 = (J * (math.exp(l) - math.exp(-l))) + U
	tmp = 0
	if l <= -5.6e+111:
		tmp = t_1
	elif l <= -1.46e+25:
		tmp = t_2
	elif l <= 0.03:
		tmp = ((l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (2.0 * J))) * t_0) + U
	elif l <= 6e+92:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_0) + U)
	t_2 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) + U)
	tmp = 0.0
	if (l <= -5.6e+111)
		tmp = t_1;
	elseif (l <= -1.46e+25)
		tmp = t_2;
	elseif (l <= 0.03)
		tmp = Float64(Float64(Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(2.0 * J))) * t_0) + U);
	elseif (l <= 6e+92)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((J * (0.3333333333333333 * (l ^ 3.0))) * t_0) + U;
	t_2 = (J * (exp(l) - exp(-l))) + U;
	tmp = 0.0;
	if (l <= -5.6e+111)
		tmp = t_1;
	elseif (l <= -1.46e+25)
		tmp = t_2;
	elseif (l <= 0.03)
		tmp = ((l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (2.0 * J))) * t_0) + U;
	elseif (l <= 6e+92)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -5.6e+111], t$95$1, If[LessEqual[l, -1.46e+25], t$95$2, If[LessEqual[l, 0.03], N[(N[(N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 6e+92], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5999999999999999e111 or 6.00000000000000026e92 < 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 99.0%

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

   4. Taylor expanded in l around inf 99.0%

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

   if -5.5999999999999999e111 < l < -1.45999999999999996e25 or 0.029999999999999999 < l < 6.00000000000000026e92

   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 92.9%

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

   if -1.45999999999999996e25 < l < 0.029999999999999999

   1. Initial program 73.5%

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

   3. Taylor expanded in l around 0 98.5%

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

Alternative 7: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_0 + U\\ t_2 := J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1950000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.00182:\\ \;\;\;\;\mathsf{fma}\left(J, \left(2 \cdot \ell\right) \cdot t\_0, U\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ (* (* J (* 0.3333333333333333 (pow l 3.0))) t_0) U))
        (t_2 (+ (* J (- (exp l) (exp (- l)))) U)))
   (if (<= l -5.6e+111)
     t_1
     (if (<= l -1950000000.0)
       t_2
       (if (<= l 0.00182)
         (fma J (* (* 2.0 l) t_0) U)
         (if (<= l 6e+92) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((J * (0.3333333333333333 * pow(l, 3.0))) * t_0) + U;
	double t_2 = (J * (exp(l) - exp(-l))) + U;
	double tmp;
	if (l <= -5.6e+111) {
		tmp = t_1;
	} else if (l <= -1950000000.0) {
		tmp = t_2;
	} else if (l <= 0.00182) {
		tmp = fma(J, ((2.0 * l) * t_0), U);
	} else if (l <= 6e+92) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_0) + U)
	t_2 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) + U)
	tmp = 0.0
	if (l <= -5.6e+111)
		tmp = t_1;
	elseif (l <= -1950000000.0)
		tmp = t_2;
	elseif (l <= 0.00182)
		tmp = fma(J, Float64(Float64(2.0 * l) * t_0), U);
	elseif (l <= 6e+92)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -5.6e+111], t$95$1, If[LessEqual[l, -1950000000.0], t$95$2, If[LessEqual[l, 0.00182], N[(J * N[(N[(2.0 * l), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 6e+92], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5999999999999999e111 or 6.00000000000000026e92 < 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 99.0%

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

   4. Taylor expanded in l around inf 99.0%

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

   if -5.5999999999999999e111 < l < -1.95e9 or 0.00182 < l < 6.00000000000000026e92

   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 93.0%

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

   if -1.95e9 < l < 0.00182

   1. Initial program 73.2%

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

      1. associate-*l* 73.2%

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

      2. fma-define 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in l around 0 98.5%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_0 + U\\ t_2 := J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{+25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.12:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ (* (* J (* 0.3333333333333333 (pow l 3.0))) t_0) U))
        (t_2 (+ (* J (- (exp l) (exp (- l)))) U)))
   (if (<= l -5.6e+111)
     t_1
     (if (<= l -1.46e+25)
       t_2
       (if (<= l 0.12)
         (+ (* (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) t_0) U)
         (if (<= l 6e+92) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((J * (0.3333333333333333 * pow(l, 3.0))) * t_0) + U;
	double t_2 = (J * (exp(l) - exp(-l))) + U;
	double tmp;
	if (l <= -5.6e+111) {
		tmp = t_1;
	} else if (l <= -1.46e+25) {
		tmp = t_2;
	} else if (l <= 0.12) {
		tmp = ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) * t_0) + U;
	} else if (l <= 6e+92) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = ((j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_0) + u
    t_2 = (j * (exp(l) - exp(-l))) + u
    if (l <= (-5.6d+111)) then
        tmp = t_1
    else if (l <= (-1.46d+25)) then
        tmp = t_2
    else if (l <= 0.12d0) then
        tmp = ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) * t_0) + u
    else if (l <= 6d+92) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_0) + U;
	double t_2 = (J * (Math.exp(l) - Math.exp(-l))) + U;
	double tmp;
	if (l <= -5.6e+111) {
		tmp = t_1;
	} else if (l <= -1.46e+25) {
		tmp = t_2;
	} else if (l <= 0.12) {
		tmp = ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) * t_0) + U;
	} else if (l <= 6e+92) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((J * (0.3333333333333333 * math.pow(l, 3.0))) * t_0) + U
	t_2 = (J * (math.exp(l) - math.exp(-l))) + U
	tmp = 0
	if l <= -5.6e+111:
		tmp = t_1
	elif l <= -1.46e+25:
		tmp = t_2
	elif l <= 0.12:
		tmp = ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) * t_0) + U
	elif l <= 6e+92:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_0) + U)
	t_2 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) + U)
	tmp = 0.0
	if (l <= -5.6e+111)
		tmp = t_1;
	elseif (l <= -1.46e+25)
		tmp = t_2;
	elseif (l <= 0.12)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) * t_0) + U);
	elseif (l <= 6e+92)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((J * (0.3333333333333333 * (l ^ 3.0))) * t_0) + U;
	t_2 = (J * (exp(l) - exp(-l))) + U;
	tmp = 0.0;
	if (l <= -5.6e+111)
		tmp = t_1;
	elseif (l <= -1.46e+25)
		tmp = t_2;
	elseif (l <= 0.12)
		tmp = ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) * t_0) + U;
	elseif (l <= 6e+92)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -5.6e+111], t$95$1, If[LessEqual[l, -1.46e+25], t$95$2, If[LessEqual[l, 0.12], N[(N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 6e+92], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5999999999999999e111 or 6.00000000000000026e92 < 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 99.0%

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

   4. Taylor expanded in l around inf 99.0%

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

   if -5.5999999999999999e111 < l < -1.45999999999999996e25 or 0.12 < l < 6.00000000000000026e92

   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 92.9%

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

   if -1.45999999999999996e25 < l < 0.12

   1. Initial program 73.5%

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

   3. Taylor expanded in l around 0 98.5%

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

Alternative 9: 80.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (* U (+ 1.0 (* 2.0 (/ (* J (* l (cos (* 0.5 K)))) U))))
   (+ (* 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((0.5 * K)))) / U)));
	} else {
		tmp = (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((0.5d0 * k)))) / u)))
    else
        tmp = (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((0.5 * K)))) / U)));
	} else {
		tmp = (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((0.5 * K)))) / U)))
	else:
		tmp = (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(Float64(J * Float64(l * cos(Float64(0.5 * K)))) / U))));
	else
		tmp = 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((0.5 * K)))) / U)));
	else
		tmp = (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[(N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $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%

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

   4. Taylor expanded in U around inf 70.0%

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

   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. Add Preprocessing

Alternative 10: 79.0% accurate, 1.4× speedup

Math

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

FPCore

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

Alternative 11: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right) + U\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -560:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-31}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))
   (if (<= l -3.9e+129)
     t_0
     (if (<= l -560.0)
       (log1p (expm1 (- (/ -4.0 U) U)))
       (if (<= l 1.5e-31) (+ (* 2.0 (* J (* l (cos (* 0.5 K))))) U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) + U;
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -560.0) {
		tmp = log1p(expm1(((-4.0 / U) - U)));
	} else if (l <= 1.5e-31) {
		tmp = (2.0 * (J * (l * cos((0.5 * K))))) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) + U;
	double tmp;
	if (l <= -3.9e+129) {
		tmp = t_0;
	} else if (l <= -560.0) {
		tmp = Math.log1p(Math.expm1(((-4.0 / U) - U)));
	} else if (l <= 1.5e-31) {
		tmp = (2.0 * (J * (l * Math.cos((0.5 * K))))) + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) + U
	tmp = 0
	if l <= -3.9e+129:
		tmp = t_0
	elif l <= -560.0:
		tmp = math.log1p(math.expm1(((-4.0 / U) - U)))
	elif l <= 1.5e-31:
		tmp = (2.0 * (J * (l * math.cos((0.5 * K))))) + U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) + U)
	tmp = 0.0
	if (l <= -3.9e+129)
		tmp = t_0;
	elseif (l <= -560.0)
		tmp = log1p(expm1(Float64(Float64(-4.0 / U) - U)));
	elseif (l <= 1.5e-31)
		tmp = Float64(Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))) + U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -3.9e+129], t$95$0, If[LessEqual[l, -560.0], N[Log[1 + N[(Exp[N[(N[(-4.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.5e-31], N[(N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8999999999999997e129 or 1.49999999999999991e-31 < l

   1. Initial program 96.9%

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

   3. Taylor expanded in l around 0 88.5%

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

   4. Taylor expanded in K around 0 75.5%

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

   if -3.8999999999999997e129 < l < -560

   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 3.1%

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

      1. log1p-expm1-u 65.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   6. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   if -560 < l < 1.49999999999999991e-31

   1. Initial program 74.5%

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

   3. Taylor expanded in l around 0 99.3%

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

Alternative 12: 57.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right) + U\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+269}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* J (* l (cos (* 0.5 K)))))))
   (if (<= l -1.5e+54)
     t_0
     (if (<= l 3.9e+30)
       (+ (* 2.0 (* J l)) U)
       (if (<= l 3.2e+132)
         (pow U -3.0)
         (if (<= l 4.4e+269) t_0 (pow U -4.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * (l * cos((0.5 * K))));
	double tmp;
	if (l <= -1.5e+54) {
		tmp = t_0;
	} else if (l <= 3.9e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 3.2e+132) {
		tmp = pow(U, -3.0);
	} else if (l <= 4.4e+269) {
		tmp = t_0;
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (j * (l * cos((0.5d0 * k))))
    if (l <= (-1.5d+54)) then
        tmp = t_0
    else if (l <= 3.9d+30) then
        tmp = (2.0d0 * (j * l)) + u
    else if (l <= 3.2d+132) then
        tmp = u ** (-3.0d0)
    else if (l <= 4.4d+269) then
        tmp = t_0
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * (l * Math.cos((0.5 * K))));
	double tmp;
	if (l <= -1.5e+54) {
		tmp = t_0;
	} else if (l <= 3.9e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 3.2e+132) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 4.4e+269) {
		tmp = t_0;
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (J * (l * math.cos((0.5 * K))))
	tmp = 0
	if l <= -1.5e+54:
		tmp = t_0
	elif l <= 3.9e+30:
		tmp = (2.0 * (J * l)) + U
	elif l <= 3.2e+132:
		tmp = math.pow(U, -3.0)
	elif l <= 4.4e+269:
		tmp = t_0
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K)))))
	tmp = 0.0
	if (l <= -1.5e+54)
		tmp = t_0;
	elseif (l <= 3.9e+30)
		tmp = Float64(Float64(2.0 * Float64(J * l)) + U);
	elseif (l <= 3.2e+132)
		tmp = U ^ -3.0;
	elseif (l <= 4.4e+269)
		tmp = t_0;
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (J * (l * cos((0.5 * K))));
	tmp = 0.0;
	if (l <= -1.5e+54)
		tmp = t_0;
	elseif (l <= 3.9e+30)
		tmp = (2.0 * (J * l)) + U;
	elseif (l <= 3.2e+132)
		tmp = U ^ -3.0;
	elseif (l <= 4.4e+269)
		tmp = t_0;
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.5e+54], t$95$0, If[LessEqual[l, 3.9e+30], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 3.2e+132], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 4.4e+269], t$95$0, N[Power[U, -4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4999999999999999e54 or 3.1999999999999997e132 < l < 4.3999999999999997e269

   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 38.5%

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

   4. Taylor expanded in J around inf 38.5%

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

   if -1.4999999999999999e54 < l < 3.90000000000000011e30

   1. Initial program 76.0%

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

   3. Taylor expanded in l around 0 88.5%

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

   4. Taylor expanded in K around 0 76.6%

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

   if 3.90000000000000011e30 < l < 3.1999999999999997e132

   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 39.4%

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

   if 4.3999999999999997e269 < 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 60.5%

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

Alternative 13: 63.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{if}\;\ell \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;t\_0 + U\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 10^{+269}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* J (* l (cos (* 0.5 K)))))))
   (if (<= l 2.45e+30)
     (+ t_0 U)
     (if (<= l 1.75e+130) (pow U -3.0) (if (<= l 1e+269) t_0 (pow U -4.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * (l * cos((0.5 * K))));
	double tmp;
	if (l <= 2.45e+30) {
		tmp = t_0 + U;
	} else if (l <= 1.75e+130) {
		tmp = pow(U, -3.0);
	} else if (l <= 1e+269) {
		tmp = t_0;
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (j * (l * cos((0.5d0 * k))))
    if (l <= 2.45d+30) then
        tmp = t_0 + u
    else if (l <= 1.75d+130) then
        tmp = u ** (-3.0d0)
    else if (l <= 1d+269) then
        tmp = t_0
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * (l * Math.cos((0.5 * K))));
	double tmp;
	if (l <= 2.45e+30) {
		tmp = t_0 + U;
	} else if (l <= 1.75e+130) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 1e+269) {
		tmp = t_0;
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (J * (l * math.cos((0.5 * K))))
	tmp = 0
	if l <= 2.45e+30:
		tmp = t_0 + U
	elif l <= 1.75e+130:
		tmp = math.pow(U, -3.0)
	elif l <= 1e+269:
		tmp = t_0
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K)))))
	tmp = 0.0
	if (l <= 2.45e+30)
		tmp = Float64(t_0 + U);
	elseif (l <= 1.75e+130)
		tmp = U ^ -3.0;
	elseif (l <= 1e+269)
		tmp = t_0;
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (J * (l * cos((0.5 * K))));
	tmp = 0.0;
	if (l <= 2.45e+30)
		tmp = t_0 + U;
	elseif (l <= 1.75e+130)
		tmp = U ^ -3.0;
	elseif (l <= 1e+269)
		tmp = t_0;
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2.45e+30], N[(t$95$0 + U), $MachinePrecision], If[LessEqual[l, 1.75e+130], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 1e+269], t$95$0, N[Power[U, -4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 2.44999999999999992e30

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0 75.5%

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

   if 2.44999999999999992e30 < l < 1.75e130

   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 39.4%

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

   if 1.75e130 < l < 1e269

   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 39.2%

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

   4. Taylor expanded in J around inf 39.3%

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

   if 1e269 < 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 60.5%

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

Alternative 14: 54.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right) + U\\ \mathbf{elif}\;\ell \leq 1.16 \cdot 10^{+127}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\ell \cdot \left(2 \cdot J + \frac{U}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.45e+30)
   (+ (* 2.0 (* J l)) U)
   (if (<= l 1.16e+127)
     (pow U -4.0)
     (if (<= l 2e+269) (* l (+ (* 2.0 J) (/ U l))) (pow U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.45e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 1.16e+127) {
		tmp = pow(U, -4.0);
	} else if (l <= 2e+269) {
		tmp = l * ((2.0 * J) + (U / l));
	} else {
		tmp = pow(U, -4.0);
	}
	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 <= 2.45d+30) then
        tmp = (2.0d0 * (j * l)) + u
    else if (l <= 1.16d+127) then
        tmp = u ** (-4.0d0)
    else if (l <= 2d+269) then
        tmp = l * ((2.0d0 * j) + (u / l))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.45e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 1.16e+127) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 2e+269) {
		tmp = l * ((2.0 * J) + (U / l));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.45e+30:
		tmp = (2.0 * (J * l)) + U
	elif l <= 1.16e+127:
		tmp = math.pow(U, -4.0)
	elif l <= 2e+269:
		tmp = l * ((2.0 * J) + (U / l))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.45e+30)
		tmp = Float64(Float64(2.0 * Float64(J * l)) + U);
	elseif (l <= 1.16e+127)
		tmp = U ^ -4.0;
	elseif (l <= 2e+269)
		tmp = Float64(l * Float64(Float64(2.0 * J) + Float64(U / l)));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.45e+30)
		tmp = (2.0 * (J * l)) + U;
	elseif (l <= 1.16e+127)
		tmp = U ^ -4.0;
	elseif (l <= 2e+269)
		tmp = l * ((2.0 * J) + (U / l));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.45e+30], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.16e+127], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 2e+269], N[(l * N[(N[(2.0 * J), $MachinePrecision] + N[(U / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.44999999999999992e30

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0 75.5%

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

   4. Taylor expanded in K around 0 63.8%

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

   if 2.44999999999999992e30 < l < 1.15999999999999994e127 or 2.0000000000000001e269 < 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 45.5%

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

   if 1.15999999999999994e127 < l < 2.0000000000000001e269

   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 38.3%

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

   4. Taylor expanded in l around inf 38.3%

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

   5. Taylor expanded in K around 0 34.9%

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

Alternative 15: 54.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \left(J \cdot \ell\right) + U\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+134}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+269}:\\ \;\;\;\;\ell \cdot \left(2 \cdot J + \frac{U}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 3.9e+30)
   (+ (* 2.0 (* J l)) U)
   (if (<= l 1.25e+134)
     (pow U -3.0)
     (if (<= l 4.4e+269) (* l (+ (* 2.0 J) (/ U l))) (pow U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.9e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 1.25e+134) {
		tmp = pow(U, -3.0);
	} else if (l <= 4.4e+269) {
		tmp = l * ((2.0 * J) + (U / l));
	} else {
		tmp = pow(U, -4.0);
	}
	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 <= 3.9d+30) then
        tmp = (2.0d0 * (j * l)) + u
    else if (l <= 1.25d+134) then
        tmp = u ** (-3.0d0)
    else if (l <= 4.4d+269) then
        tmp = l * ((2.0d0 * j) + (u / l))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.9e+30) {
		tmp = (2.0 * (J * l)) + U;
	} else if (l <= 1.25e+134) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 4.4e+269) {
		tmp = l * ((2.0 * J) + (U / l));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 3.9e+30:
		tmp = (2.0 * (J * l)) + U
	elif l <= 1.25e+134:
		tmp = math.pow(U, -3.0)
	elif l <= 4.4e+269:
		tmp = l * ((2.0 * J) + (U / l))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 3.9e+30)
		tmp = Float64(Float64(2.0 * Float64(J * l)) + U);
	elseif (l <= 1.25e+134)
		tmp = U ^ -3.0;
	elseif (l <= 4.4e+269)
		tmp = Float64(l * Float64(Float64(2.0 * J) + Float64(U / l)));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 3.9e+30)
		tmp = (2.0 * (J * l)) + U;
	elseif (l <= 1.25e+134)
		tmp = U ^ -3.0;
	elseif (l <= 4.4e+269)
		tmp = l * ((2.0 * J) + (U / l));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 3.9e+30], N[(N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.25e+134], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 4.4e+269], N[(l * N[(N[(2.0 * J), $MachinePrecision] + N[(U / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.90000000000000011e30

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0 75.5%

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

   4. Taylor expanded in K around 0 63.8%

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

   if 3.90000000000000011e30 < l < 1.24999999999999995e134

   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 39.4%

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

   if 1.24999999999999995e134 < l < 4.3999999999999997e269

   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 39.2%

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

   4. Taylor expanded in l around inf 39.2%

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

   5. Taylor expanded in K around 0 35.8%

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

   if 4.3999999999999997e269 < 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 60.5%

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

Alternative 16: 41.6% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+48}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -9.2e+48) (* U U) (if (<= l 1.7e-9) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -9.2e+48) {
		tmp = U * U;
	} else if (l <= 1.7e-9) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-9.2d+48)) then
        tmp = u * u
    else if (l <= 1.7d-9) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -9.2e+48) {
		tmp = U * U;
	} else if (l <= 1.7e-9) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -9.2e+48:
		tmp = U * U
	elif l <= 1.7e-9:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -9.2e+48)
		tmp = Float64(U * U);
	elseif (l <= 1.7e-9)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -9.2e+48)
		tmp = U * U;
	elseif (l <= 1.7e-9)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -9.2e+48], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.7e-9], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.2000000000000001e48 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 -9.2000000000000001e48 < 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. Add Preprocessing

Alternative 17: 42.0% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -950:\\ \;\;\;\;-4 + \left(-U\right) \cdot U\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -950.0) (+ -4.0 (* (- U) U)) (if (<= l 1.7e-9) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -950.0) {
		tmp = -4.0 + (-U * U);
	} else if (l <= 1.7e-9) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-950.0d0)) then
        tmp = (-4.0d0) + (-u * u)
    else if (l <= 1.7d-9) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -950.0) {
		tmp = -4.0 + (-U * U);
	} else if (l <= 1.7e-9) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -950.0:
		tmp = -4.0 + (-U * U)
	elif l <= 1.7e-9:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -950.0)
		tmp = Float64(-4.0 + Float64(Float64(-U) * U));
	elseif (l <= 1.7e-9)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -950.0)
		tmp = -4.0 + (-U * U);
	elseif (l <= 1.7e-9)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -950.0], N[(-4.0 + N[((-U) * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e-9], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -950

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 17.4%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]

   if -950 < l < 1.6999999999999999e-9

   1. Initial program 73.3%

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

   3. Taylor expanded in J around 0 72.1%

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

   if 1.6999999999999999e-9 < 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 13.3%

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

Alternative 18: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in K around 0 51.4%

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

Alternative 19: 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. Add Preprocessing

Reproduce

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