Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.5%

Time: 18.6s

Alternatives: 16

Speedup: 1.4×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       (* J (cos (* K 0.5)))
       (fma 2.0 l (* 0.3333333333333333 (pow l 3.0))))))))

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 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((J * cos((K * 0.5))) * fma(2.0, l, (0.3333333333333333 * pow(l, 3.0))));
	}
	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 <= 0.0))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * fma(2.0, l, Float64(0.3333333333333333 * (l ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * l + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $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 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 73.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   3. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)} + U \]

      2. +-commutative 99.9%

         \[\leadsto \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

      3. fma-def 99.9%

         \[\leadsto \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

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 <= -((double) INFINITY)) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.0):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	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 <= Float64(-Inf)) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.0)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	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[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $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 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 73.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -0.0001 \lor \neg \left(t_0 \leq 1\right):\\ \;\;\;\;t_0 \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.0001) (not (<= t_0 1.0)))
     (+ (* t_0 J) U)
     (+ U (* (cos (/ K 2.0)) (* J (* l 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 <= -0.0001) || !(t_0 <= 1.0)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-0.0001d0)) .or. (.not. (t_0 <= 1.0d0))) then
        tmp = (t_0 * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

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 <= -0.0001) || !(t_0 <= 1.0)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.0001) or not (t_0 <= 1.0):
		tmp = (t_0 * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 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 <= -0.0001) || !(t_0 <= 1.0))
		tmp = Float64(Float64(t_0 * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 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 <= -0.0001) || ~((t_0 <= 1.0)))
		tmp = (t_0 * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 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, -0.0001], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1.00000000000000005e-4 or 1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 69.6%

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

   if -1.00000000000000005e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1

   1. Initial program 74.0%

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

   2. Taylor expanded in l around 0 99.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.0001 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 1\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 4: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{if}\;t_0 \leq -0.535:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.35:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K)))))))))
   (if (<= t_0 -0.535)
     t_1
     (if (<= t_0 -0.35)
       (+ U (* t_0 (* J (* l 2.0))))
       (if (<= t_0 -0.02)
         t_1
         (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double tmp;
	if (t_0 <= -0.535) {
		tmp = t_1;
	} else if (t_0 <= -0.35) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (2.0d0 * (j * (l + (l * ((-0.125d0) * (k * k))))))
    if (t_0 <= (-0.535d0)) then
        tmp = t_1
    else if (t_0 <= (-0.35d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else if (t_0 <= (-0.02d0)) then
        tmp = t_1
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    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 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double tmp;
	if (t_0 <= -0.535) {
		tmp = t_1;
	} else if (t_0 <= -0.35) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))))
	tmp = 0
	if t_0 <= -0.535:
		tmp = t_1
	elif t_0 <= -0.35:
		tmp = U + (t_0 * (J * (l * 2.0)))
	elif t_0 <= -0.02:
		tmp = t_1
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))))
	tmp = 0.0
	if (t_0 <= -0.535)
		tmp = t_1;
	elseif (t_0 <= -0.35)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	tmp = 0.0;
	if (t_0 <= -0.535)
		tmp = t_1;
	elseif (t_0 <= -0.35)
		tmp = U + (t_0 * (J * (l * 2.0)));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	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[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.535], t$95$1, If[LessEqual[t$95$0, -0.35], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.535000000000000031 or -0.34999999999999998 < (cos.f64 (/.f64 K 2)) < -0.0200000000000000004

   1. Initial program 90.6%

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

   2. Taylor expanded in l around 0 48.8%

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

   3. Taylor expanded in K around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. unpow2 67.2%

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

   5. Simplified 67.2%

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

   if -0.535000000000000031 < (cos.f64 (/.f64 K 2)) < -0.34999999999999998

   1. Initial program 82.1%

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

   2. Taylor expanded in l around 0 65.6%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0 88.8%

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

   3. Taylor expanded in K around 0 84.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.535:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.35:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + 0.3333333333333333 \cdot \left(t_0 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.035:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* 0.3333333333333333 (* t_0 (* J (pow l 3.0))))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1e+77)
     t_1
     (if (<= l -0.035)
       t_2
       (if (<= l 15500.0)
         (+ U (* J (* t_0 (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 2.2e+100) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (0.3333333333333333 * (t_0 * (J * pow(l, 3.0))));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_1;
	} else if (l <= -0.035) {
		tmp = t_2;
	} else if (l <= 15500.0) {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.2e+100) {
		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 * 0.5d0))
    t_1 = u + (0.3333333333333333d0 * (t_0 * (j * (l ** 3.0d0))))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1d+77)) then
        tmp = t_1
    else if (l <= (-0.035d0)) then
        tmp = t_2
    else if (l <= 15500.0d0) then
        tmp = u + (j * (t_0 * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 2.2d+100) 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 * 0.5));
	double t_1 = U + (0.3333333333333333 * (t_0 * (J * Math.pow(l, 3.0))));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_1;
	} else if (l <= -0.035) {
		tmp = t_2;
	} else if (l <= 15500.0) {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.2e+100) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (0.3333333333333333 * (t_0 * (J * math.pow(l, 3.0))))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1e+77:
		tmp = t_1
	elif l <= -0.035:
		tmp = t_2
	elif l <= 15500.0:
		tmp = U + (J * (t_0 * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 2.2e+100:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(0.3333333333333333 * Float64(t_0 * Float64(J * (l ^ 3.0)))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1e+77)
		tmp = t_1;
	elseif (l <= -0.035)
		tmp = t_2;
	elseif (l <= 15500.0)
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 2.2e+100)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + (0.3333333333333333 * (t_0 * (J * (l ^ 3.0))));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1e+77)
		tmp = t_1;
	elseif (l <= -0.035)
		tmp = t_2;
	elseif (l <= 15500.0)
		tmp = U + (J * (t_0 * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 2.2e+100)
		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 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.3333333333333333 * N[(t$95$0 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1e+77], t$95$1, If[LessEqual[l, -0.035], t$95$2, If[LessEqual[l, 15500.0], N[(U + N[(J * N[(t$95$0 * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+100], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999983e76 or 2.2000000000000001e100 < 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. Taylor expanded in l around 0 95.2%

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

   3. Taylor expanded in l around inf 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -9.99999999999999983e76 < l < -0.035000000000000003 or 15500 < l < 2.2000000000000001e100

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 75.8%

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

   if -0.035000000000000003 < l < 15500

   1. Initial program 74.3%

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

   2. Taylor expanded in l around 0 98.6%

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

   3. Taylor expanded in J around 0 98.6%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.035:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 6: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.23:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.3333333333333333 (* (cos (* K 0.5)) (* J (pow l 3.0))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1e+77)
     t_0
     (if (<= l -0.23)
       t_1
       (if (<= l 15500.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 1.8e+100) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (cos((K * 0.5)) * (J * pow(l, 3.0))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_0;
	} else if (l <= -0.23) {
		tmp = t_1;
	} else if (l <= 15500.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.8e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (cos((k * 0.5d0)) * (j * (l ** 3.0d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1d+77)) then
        tmp = t_0
    else if (l <= (-0.23d0)) then
        tmp = t_1
    else if (l <= 15500.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 1.8d+100) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (Math.cos((K * 0.5)) * (J * Math.pow(l, 3.0))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_0;
	} else if (l <= -0.23) {
		tmp = t_1;
	} else if (l <= 15500.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.8e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (math.cos((K * 0.5)) * (J * math.pow(l, 3.0))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1e+77:
		tmp = t_0
	elif l <= -0.23:
		tmp = t_1
	elif l <= 15500.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 1.8e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(cos(Float64(K * 0.5)) * Float64(J * (l ^ 3.0)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1e+77)
		tmp = t_0;
	elseif (l <= -0.23)
		tmp = t_1;
	elseif (l <= 15500.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 1.8e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (cos((K * 0.5)) * (J * (l ^ 3.0))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1e+77)
		tmp = t_0;
	elseif (l <= -0.23)
		tmp = t_1;
	elseif (l <= 15500.0)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 1.8e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1e+77], t$95$0, If[LessEqual[l, -0.23], t$95$1, If[LessEqual[l, 15500.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999983e76 or 1.8e100 < 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. Taylor expanded in l around 0 95.2%

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

   3. Taylor expanded in l around inf 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -9.99999999999999983e76 < l < -0.23000000000000001 or 15500 < l < 1.8e100

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 75.8%

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

   if -0.23000000000000001 < l < 15500

   1. Initial program 74.3%

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

   2. Taylor expanded in l around 0 98.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.23:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 7: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.3333333333333333 (* (cos (* K 0.5)) (* J (pow l 3.0))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1e+77)
     t_0
     (if (<= l -8.4e-5)
       t_1
       (if (<= l 15500.0)
         (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
         (if (<= l 1.8e+100) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (cos((K * 0.5)) * (J * pow(l, 3.0))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_0;
	} else if (l <= -8.4e-5) {
		tmp = t_1;
	} else if (l <= 15500.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.8e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (cos((k * 0.5d0)) * (j * (l ** 3.0d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1d+77)) then
        tmp = t_0
    else if (l <= (-8.4d-5)) then
        tmp = t_1
    else if (l <= 15500.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 1.8d+100) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (Math.cos((K * 0.5)) * (J * Math.pow(l, 3.0))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1e+77) {
		tmp = t_0;
	} else if (l <= -8.4e-5) {
		tmp = t_1;
	} else if (l <= 15500.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.8e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (math.cos((K * 0.5)) * (J * math.pow(l, 3.0))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1e+77:
		tmp = t_0
	elif l <= -8.4e-5:
		tmp = t_1
	elif l <= 15500.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 1.8e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(cos(Float64(K * 0.5)) * Float64(J * (l ^ 3.0)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1e+77)
		tmp = t_0;
	elseif (l <= -8.4e-5)
		tmp = t_1;
	elseif (l <= 15500.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.8e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (cos((K * 0.5)) * (J * (l ^ 3.0))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1e+77)
		tmp = t_0;
	elseif (l <= -8.4e-5)
		tmp = t_1;
	elseif (l <= 15500.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 1.8e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1e+77], t$95$0, If[LessEqual[l, -8.4e-5], t$95$1, If[LessEqual[l, 15500.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999983e76 or 1.8e100 < 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. Taylor expanded in l around 0 95.2%

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

   3. Taylor expanded in l around inf 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -9.99999999999999983e76 < l < -8.39999999999999954e-5 or 15500 < l < 1.8e100

   1. Initial program 99.6%

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

   2. Taylor expanded in K around 0 76.0%

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

   if -8.39999999999999954e-5 < l < 15500

   1. Initial program 74.2%

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

   2. Taylor expanded in l around 0 98.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 15500:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 8: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K)))))))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + (2.0d0 * (j * (l + (l * ((-0.125d0) * (k * k))))))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0200000000000000004

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 50.9%

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

   3. Taylor expanded in K around 0 60.5%

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

      1. associate-*r* 60.5%

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

      2. unpow2 60.5%

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

   5. Simplified 60.5%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0 88.8%

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

   3. Taylor expanded in l around inf 74.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 74.0%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 74.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 9: 77.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ t_1 := U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ t_2 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -50000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 190000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+200} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (+ (* (* K K) -64.0) 512.0))))
        (t_1 (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K))))))))
        (t_2 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -1.4e+101)
     t_2
     (if (<= l -6e+77)
       t_1
       (if (<= l -1.25e+41)
         t_0
         (if (<= l -50000000000.0)
           (* U U)
           (if (<= l -8.4e-5)
             t_0
             (if (<= l 190000.0)
               (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
               (if (or (<= l 1.45e+200) (not (<= l 5e+251))) t_2 t_1)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (((K * K) * -64.0) + 512.0));
	double t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double t_2 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -1.4e+101) {
		tmp = t_2;
	} else if (l <= -6e+77) {
		tmp = t_1;
	} else if (l <= -1.25e+41) {
		tmp = t_0;
	} else if (l <= -50000000000.0) {
		tmp = U * U;
	} else if (l <= -8.4e-5) {
		tmp = t_0;
	} else if (l <= 190000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if ((l <= 1.45e+200) || !(l <= 5e+251)) {
		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 = u + (j * (((k * k) * (-64.0d0)) + 512.0d0))
    t_1 = u + (2.0d0 * (j * (l + (l * ((-0.125d0) * (k * k))))))
    t_2 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-1.4d+101)) then
        tmp = t_2
    else if (l <= (-6d+77)) then
        tmp = t_1
    else if (l <= (-1.25d+41)) then
        tmp = t_0
    else if (l <= (-50000000000.0d0)) then
        tmp = u * u
    else if (l <= (-8.4d-5)) then
        tmp = t_0
    else if (l <= 190000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if ((l <= 1.45d+200) .or. (.not. (l <= 5d+251))) 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 = U + (J * (((K * K) * -64.0) + 512.0));
	double t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double t_2 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -1.4e+101) {
		tmp = t_2;
	} else if (l <= -6e+77) {
		tmp = t_1;
	} else if (l <= -1.25e+41) {
		tmp = t_0;
	} else if (l <= -50000000000.0) {
		tmp = U * U;
	} else if (l <= -8.4e-5) {
		tmp = t_0;
	} else if (l <= 190000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if ((l <= 1.45e+200) || !(l <= 5e+251)) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (((K * K) * -64.0) + 512.0))
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))))
	t_2 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -1.4e+101:
		tmp = t_2
	elif l <= -6e+77:
		tmp = t_1
	elif l <= -1.25e+41:
		tmp = t_0
	elif l <= -50000000000.0:
		tmp = U * U
	elif l <= -8.4e-5:
		tmp = t_0
	elif l <= 190000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif (l <= 1.45e+200) or not (l <= 5e+251):
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * -64.0) + 512.0)))
	t_1 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))))
	t_2 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -1.4e+101)
		tmp = t_2;
	elseif (l <= -6e+77)
		tmp = t_1;
	elseif (l <= -1.25e+41)
		tmp = t_0;
	elseif (l <= -50000000000.0)
		tmp = Float64(U * U);
	elseif (l <= -8.4e-5)
		tmp = t_0;
	elseif (l <= 190000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif ((l <= 1.45e+200) || !(l <= 5e+251))
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (((K * K) * -64.0) + 512.0));
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	t_2 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -1.4e+101)
		tmp = t_2;
	elseif (l <= -6e+77)
		tmp = t_1;
	elseif (l <= -1.25e+41)
		tmp = t_0;
	elseif (l <= -50000000000.0)
		tmp = U * U;
	elseif (l <= -8.4e-5)
		tmp = t_0;
	elseif (l <= 190000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif ((l <= 1.45e+200) || ~((l <= 5e+251)))
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * -64.0), $MachinePrecision] + 512.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+101], t$95$2, If[LessEqual[l, -6e+77], t$95$1, If[LessEqual[l, -1.25e+41], t$95$0, If[LessEqual[l, -50000000000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, -8.4e-5], t$95$0, If[LessEqual[l, 190000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 1.45e+200], N[Not[LessEqual[l, 5e+251]], $MachinePrecision]], t$95$2, t$95$1]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.39999999999999991e101 or 1.9e5 < l < 1.4499999999999999e200 or 5.0000000000000005e251 < 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. Taylor expanded in l around 0 86.1%

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

   3. Taylor expanded in l around inf 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 86.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 66.5%

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

   if -1.39999999999999991e101 < l < -5.9999999999999996e77 or 1.4499999999999999e200 < l < 5.0000000000000005e251

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 24.5%

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

   3. Taylor expanded in K around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. unpow2 87.1%

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

   5. Simplified 87.1%

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

   if -5.9999999999999996e77 < l < -1.25000000000000006e41 or -5e10 < l < -8.39999999999999954e-5

   1. Initial program 98.7%

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

   3. Applied egg-rr 2.2%

      \[\leadsto \left(J \cdot \color{blue}{512}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 46.7%

      \[\leadsto \color{blue}{\left(-64 \cdot \left(J \cdot {K}^{2}\right) + 512 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot -64} + 512 \cdot J\right) + U \]

      2. associate-*l* 46.7%

         \[\leadsto \left(\color{blue}{J \cdot \left({K}^{2} \cdot -64\right)} + 512 \cdot J\right) + U \]

      3. *-commutative 46.7%

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot -64\right) + \color{blue}{J \cdot 512}\right) + U \]

      4. distribute-lft-out 46.7%

         \[\leadsto \color{blue}{J \cdot \left({K}^{2} \cdot -64 + 512\right)} + U \]

      5. unpow2 46.7%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot -64 + 512\right) + U \]

   6. Simplified 46.7%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)} + U \]

   if -1.25000000000000006e41 < l < -5e10

   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-def 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%

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

   5. Applied egg-rr 38.5%

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

   if -8.39999999999999954e-5 < l < 1.9e5

   1. Initial program 74.2%

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

   2. Taylor expanded in l around 0 98.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+101}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+77}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq -50000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq 190000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+200} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ t_1 := U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ t_2 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -17000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 29000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+197} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (+ (* (* K K) -64.0) 512.0))))
        (t_1 (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K))))))))
        (t_2 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -3.2e+103)
     t_2
     (if (<= l -4.4e+77)
       t_1
       (if (<= l -1.5e+39)
         t_0
         (if (<= l -17000000000.0)
           (* U U)
           (if (<= l -8.4e-5)
             t_0
             (if (<= l 29000.0)
               (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
               (if (or (<= l 4.9e+197) (not (<= l 5e+251))) t_2 t_1)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (((K * K) * -64.0) + 512.0));
	double t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double t_2 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -3.2e+103) {
		tmp = t_2;
	} else if (l <= -4.4e+77) {
		tmp = t_1;
	} else if (l <= -1.5e+39) {
		tmp = t_0;
	} else if (l <= -17000000000.0) {
		tmp = U * U;
	} else if (l <= -8.4e-5) {
		tmp = t_0;
	} else if (l <= 29000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if ((l <= 4.9e+197) || !(l <= 5e+251)) {
		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 = u + (j * (((k * k) * (-64.0d0)) + 512.0d0))
    t_1 = u + (2.0d0 * (j * (l + (l * ((-0.125d0) * (k * k))))))
    t_2 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-3.2d+103)) then
        tmp = t_2
    else if (l <= (-4.4d+77)) then
        tmp = t_1
    else if (l <= (-1.5d+39)) then
        tmp = t_0
    else if (l <= (-17000000000.0d0)) then
        tmp = u * u
    else if (l <= (-8.4d-5)) then
        tmp = t_0
    else if (l <= 29000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if ((l <= 4.9d+197) .or. (.not. (l <= 5d+251))) 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 = U + (J * (((K * K) * -64.0) + 512.0));
	double t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double t_2 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -3.2e+103) {
		tmp = t_2;
	} else if (l <= -4.4e+77) {
		tmp = t_1;
	} else if (l <= -1.5e+39) {
		tmp = t_0;
	} else if (l <= -17000000000.0) {
		tmp = U * U;
	} else if (l <= -8.4e-5) {
		tmp = t_0;
	} else if (l <= 29000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if ((l <= 4.9e+197) || !(l <= 5e+251)) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (((K * K) * -64.0) + 512.0))
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))))
	t_2 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -3.2e+103:
		tmp = t_2
	elif l <= -4.4e+77:
		tmp = t_1
	elif l <= -1.5e+39:
		tmp = t_0
	elif l <= -17000000000.0:
		tmp = U * U
	elif l <= -8.4e-5:
		tmp = t_0
	elif l <= 29000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif (l <= 4.9e+197) or not (l <= 5e+251):
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * -64.0) + 512.0)))
	t_1 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))))
	t_2 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -3.2e+103)
		tmp = t_2;
	elseif (l <= -4.4e+77)
		tmp = t_1;
	elseif (l <= -1.5e+39)
		tmp = t_0;
	elseif (l <= -17000000000.0)
		tmp = Float64(U * U);
	elseif (l <= -8.4e-5)
		tmp = t_0;
	elseif (l <= 29000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif ((l <= 4.9e+197) || !(l <= 5e+251))
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (((K * K) * -64.0) + 512.0));
	t_1 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	t_2 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -3.2e+103)
		tmp = t_2;
	elseif (l <= -4.4e+77)
		tmp = t_1;
	elseif (l <= -1.5e+39)
		tmp = t_0;
	elseif (l <= -17000000000.0)
		tmp = U * U;
	elseif (l <= -8.4e-5)
		tmp = t_0;
	elseif (l <= 29000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif ((l <= 4.9e+197) || ~((l <= 5e+251)))
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * -64.0), $MachinePrecision] + 512.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.2e+103], t$95$2, If[LessEqual[l, -4.4e+77], t$95$1, If[LessEqual[l, -1.5e+39], t$95$0, If[LessEqual[l, -17000000000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, -8.4e-5], t$95$0, If[LessEqual[l, 29000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.9e+197], N[Not[LessEqual[l, 5e+251]], $MachinePrecision]], t$95$2, t$95$1]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -3.19999999999999993e103 or 29000 < l < 4.90000000000000026e197 or 5.0000000000000005e251 < 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. Taylor expanded in l around 0 86.1%

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

   3. Taylor expanded in l around inf 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 86.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 66.5%

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

   if -3.19999999999999993e103 < l < -4.4000000000000001e77 or 4.90000000000000026e197 < l < 5.0000000000000005e251

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 24.5%

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

   3. Taylor expanded in K around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. unpow2 87.1%

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

   5. Simplified 87.1%

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

   if -4.4000000000000001e77 < l < -1.5e39 or -1.7e10 < l < -8.39999999999999954e-5

   1. Initial program 98.7%

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

   3. Applied egg-rr 2.2%

      \[\leadsto \left(J \cdot \color{blue}{512}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 46.7%

      \[\leadsto \color{blue}{\left(-64 \cdot \left(J \cdot {K}^{2}\right) + 512 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot -64} + 512 \cdot J\right) + U \]

      2. associate-*l* 46.7%

         \[\leadsto \left(\color{blue}{J \cdot \left({K}^{2} \cdot -64\right)} + 512 \cdot J\right) + U \]

      3. *-commutative 46.7%

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot -64\right) + \color{blue}{J \cdot 512}\right) + U \]

      4. distribute-lft-out 46.7%

         \[\leadsto \color{blue}{J \cdot \left({K}^{2} \cdot -64 + 512\right)} + U \]

      5. unpow2 46.7%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot -64 + 512\right) + U \]

   6. Simplified 46.7%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)} + U \]

   if -1.5e39 < l < -1.7e10

   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-def 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%

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

   5. Applied egg-rr 38.5%

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

   if -8.39999999999999954e-5 < l < 29000

   1. Initial program 74.2%

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

   2. Taylor expanded in l around 0 98.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+103}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq -17000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq 29000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+197} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ t_1 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+39}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq -50000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U + \mathsf{fma}\left(J, 0.125, \left(K \cdot K\right) \cdot \left(J \cdot -0.015625\right)\right)\\ \mathbf{elif}\;\ell \leq 210000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+197} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K))))))))
        (t_1 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -3.4e+105)
     t_1
     (if (<= l -4e+77)
       t_0
       (if (<= l -1.25e+39)
         (+ U (* J (+ (* (* K K) -64.0) 512.0)))
         (if (<= l -50000000000.0)
           (* U U)
           (if (<= l -8.4e-5)
             (+ U (fma J 0.125 (* (* K K) (* J -0.015625))))
             (if (<= l 210000.0)
               (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
               (if (or (<= l 4.9e+197) (not (<= l 5e+251))) t_1 t_0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	double t_1 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -3.4e+105) {
		tmp = t_1;
	} else if (l <= -4e+77) {
		tmp = t_0;
	} else if (l <= -1.25e+39) {
		tmp = U + (J * (((K * K) * -64.0) + 512.0));
	} else if (l <= -50000000000.0) {
		tmp = U * U;
	} else if (l <= -8.4e-5) {
		tmp = U + fma(J, 0.125, ((K * K) * (J * -0.015625)));
	} else if (l <= 210000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if ((l <= 4.9e+197) || !(l <= 5e+251)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))))
	t_1 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -3.4e+105)
		tmp = t_1;
	elseif (l <= -4e+77)
		tmp = t_0;
	elseif (l <= -1.25e+39)
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * -64.0) + 512.0)));
	elseif (l <= -50000000000.0)
		tmp = Float64(U * U);
	elseif (l <= -8.4e-5)
		tmp = Float64(U + fma(J, 0.125, Float64(Float64(K * K) * Float64(J * -0.015625))));
	elseif (l <= 210000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif ((l <= 4.9e+197) || !(l <= 5e+251))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+105], t$95$1, If[LessEqual[l, -4e+77], t$95$0, If[LessEqual[l, -1.25e+39], N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * -64.0), $MachinePrecision] + 512.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -50000000000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, -8.4e-5], N[(U + N[(J * 0.125 + N[(N[(K * K), $MachinePrecision] * N[(J * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 210000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.9e+197], N[Not[LessEqual[l, 5e+251]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -3.3999999999999999e105 or 2.1e5 < l < 4.90000000000000026e197 or 5.0000000000000005e251 < 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. Taylor expanded in l around 0 86.1%

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

   3. Taylor expanded in l around inf 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 86.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 66.5%

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

   if -3.3999999999999999e105 < l < -3.99999999999999993e77 or 4.90000000000000026e197 < l < 5.0000000000000005e251

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 24.5%

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

   3. Taylor expanded in K around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. unpow2 87.1%

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

   5. Simplified 87.1%

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

   if -3.99999999999999993e77 < l < -1.25000000000000004e39

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.7%

      \[\leadsto \left(J \cdot \color{blue}{512}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 51.4%

      \[\leadsto \color{blue}{\left(-64 \cdot \left(J \cdot {K}^{2}\right) + 512 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. *-commutative 51.4%

         \[\leadsto \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot -64} + 512 \cdot J\right) + U \]

      2. associate-*l* 51.4%

         \[\leadsto \left(\color{blue}{J \cdot \left({K}^{2} \cdot -64\right)} + 512 \cdot J\right) + U \]

      3. *-commutative 51.4%

         \[\leadsto \left(J \cdot \left({K}^{2} \cdot -64\right) + \color{blue}{J \cdot 512}\right) + U \]

      4. distribute-lft-out 51.4%

         \[\leadsto \color{blue}{J \cdot \left({K}^{2} \cdot -64 + 512\right)} + U \]

      5. unpow2 51.4%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot -64 + 512\right) + U \]

   6. Simplified 51.4%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)} + U \]

   if -1.25000000000000004e39 < l < -5e10

   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-def 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%

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

   5. Applied egg-rr 38.5%

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

   if -5e10 < l < -8.39999999999999954e-5

   1. Initial program 95.1%

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

   3. Applied egg-rr 0.8%

      \[\leadsto \left(J \cdot \color{blue}{0.125}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 34.1%

      \[\leadsto \color{blue}{\left(-0.015625 \cdot \left(J \cdot {K}^{2}\right) + 0.125 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. +-commutative 34.1%

         \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]

      2. *-commutative 34.1%

         \[\leadsto \left(\color{blue}{J \cdot 0.125} + -0.015625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      3. fma-def 34.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, 0.125, -0.015625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]

      4. *-commutative 34.1%

         \[\leadsto \mathsf{fma}\left(J, 0.125, \color{blue}{\left(J \cdot {K}^{2}\right) \cdot -0.015625}\right) + U \]

      5. *-commutative 34.1%

         \[\leadsto \mathsf{fma}\left(J, 0.125, \color{blue}{\left({K}^{2} \cdot J\right)} \cdot -0.015625\right) + U \]

      6. associate-*l* 34.1%

         \[\leadsto \mathsf{fma}\left(J, 0.125, \color{blue}{{K}^{2} \cdot \left(J \cdot -0.015625\right)}\right) + U \]

      7. unpow2 34.1%

         \[\leadsto \mathsf{fma}\left(J, 0.125, \color{blue}{\left(K \cdot K\right)} \cdot \left(J \cdot -0.015625\right)\right) + U \]

   6. Simplified 34.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, 0.125, \left(K \cdot K\right) \cdot \left(J \cdot -0.015625\right)\right)} + U \]

   if -8.39999999999999954e-5 < l < 2.1e5

   1. Initial program 74.2%

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

   2. Taylor expanded in l around 0 98.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+105}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{+77}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+39}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot -64 + 512\right)\\ \mathbf{elif}\;\ell \leq -50000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U + \mathsf{fma}\left(J, 0.125, \left(K \cdot K\right) \cdot \left(J \cdot -0.015625\right)\right)\\ \mathbf{elif}\;\ell \leq 210000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+197} \lor \neg \left(\ell \leq 5 \cdot 10^{+251}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]

Alternative 12: 61.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+20} \lor \neg \left(\ell \leq 75\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1e+20) (not (<= l 75.0)))
   (+ U (* 2.0 (* J (+ l (* l (* -0.125 (* K K)))))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1e+20) || !(l <= 75.0)) {
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1d+20)) .or. (.not. (l <= 75.0d0))) then
        tmp = u + (2.0d0 * (j * (l + (l * ((-0.125d0) * (k * k))))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1e+20) || !(l <= 75.0)) {
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1e+20) or not (l <= 75.0):
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1e+20) || !(l <= 75.0))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(l * Float64(-0.125 * Float64(K * K)))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1e+20) || ~((l <= 75.0)))
		tmp = U + (2.0 * (J * (l + (l * (-0.125 * (K * K))))));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1e+20], N[Not[LessEqual[l, 75.0]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l + N[(l * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1e20 or 75 < 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. Taylor expanded in l around 0 26.5%

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

   3. Taylor expanded in K around 0 43.2%

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

      1. associate-*r* 43.2%

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

      2. unpow2 43.2%

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

   5. Simplified 43.2%

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

   if -1e20 < l < 75

   1. Initial program 75.3%

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

   2. Taylor expanded in K around 0 72.2%

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

   3. Taylor expanded in l around 0 82.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+20} \lor \neg \left(\ell \leq 75\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 13: 41.9% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 0.38:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.4e-5) (* U U) (if (<= l 0.38) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.4e-5) {
		tmp = U * U;
	} else if (l <= 0.38) {
		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 <= (-8.4d-5)) then
        tmp = u * u
    else if (l <= 0.38d0) 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 <= -8.4e-5) {
		tmp = U * U;
	} else if (l <= 0.38) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.4e-5:
		tmp = U * U
	elif l <= 0.38:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.4e-5)
		tmp = Float64(U * U);
	elseif (l <= 0.38)
		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 <= -8.4e-5)
		tmp = U * U;
	elseif (l <= 0.38)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.4e-5], N[(U * U), $MachinePrecision], If[LessEqual[l, 0.38], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -8.39999999999999954e-5 or 0.38 < l

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 18.1%

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

   if -8.39999999999999954e-5 < l < 0.38

   1. Initial program 73.8%

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

      1. associate-*l* 73.8%

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

      2. fma-def 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in J around 0 71.2%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 0.38:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 14: 55.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

2. Taylor expanded in K around 0 70.5%

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

3. Taylor expanded in l around 0 52.9%

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

4. Final simplification 52.9%

   \[\leadsto U + J \cdot \left(\ell \cdot 2\right) \]

Alternative 15: 2.7% 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 87.4%

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

   1. associate-*l* 87.4%

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

   2. fma-def 87.4%

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

3. Simplified 87.4%

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

5. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{\frac{-8 - U}{-8 - U}} \]
6. Step-by-step derivation

   1. *-inverses 3.0%

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

7. Simplified 3.0%

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

8. Final simplification 3.0%

   \[\leadsto 1 \]

Alternative 16: 37.2% 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 87.4%

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

   1. associate-*l* 87.4%

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

   2. fma-def 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in J around 0 35.4%

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

5. Final simplification 35.4%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023285 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))