Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.8%

Time: 11.9s

Alternatives: 14

Speedup: 1.4×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.0002\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 0.0002)))
     (+ (* t_0 (* t_1 J)) 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 <= 0.0002)) {
		tmp = (t_0 * (t_1 * J)) + 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 <= 0.0002)) {
		tmp = (t_0 * (t_1 * J)) + 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 <= 0.0002):
		tmp = (t_0 * (t_1 * J)) + 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 <= 0.0002))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + 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 <= 0.0002)))
		tmp = (t_0 * (t_1 * J)) + 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, 0.0002]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $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 2.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.2%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-4

   1. Initial program 78.4%

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

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0002\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;U + t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0002)))
     (+ U (* t_0 J))
     (+ U (* (cos (/ K 2.0)) (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0002)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0002)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0002):
		tmp = U + (t_0 * J)
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0002))
		tmp = Float64(U + Float64(t_0 * J));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0002)))
		tmp = U + (t_0 * J);
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0002]], $MachinePrecision]], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.2%

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

   2. Taylor expanded in K around 0 70.3%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-4

   1. Initial program 78.4%

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

   2. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0002\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0245 \lor \neg \left(\ell \leq 46\right) \land \ell \leq 8 \cdot 10^{+91}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -9.2e+132)
     (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
     (if (or (<= l -0.0245) (and (not (<= l 46.0)) (<= l 8e+91)))
       (+ U (* (- (exp l) (exp (- l))) (+ J (* J (* K (* K -0.125))))))
       (+ 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 tmp;
	if (l <= -9.2e+132) {
		tmp = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	} else if ((l <= -0.0245) || (!(l <= 46.0) && (l <= 8e+91))) {
		tmp = U + ((exp(l) - exp(-l)) * (J + (J * (K * (K * -0.125)))));
	} else {
		tmp = U + (t_0 * (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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-9.2d+132)) then
        tmp = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    else if ((l <= (-0.0245d0)) .or. (.not. (l <= 46.0d0)) .and. (l <= 8d+91)) then
        tmp = u + ((exp(l) - exp(-l)) * (j + (j * (k * (k * (-0.125d0))))))
    else
        tmp = u + (t_0 * (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 tmp;
	if (l <= -9.2e+132) {
		tmp = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	} else if ((l <= -0.0245) || (!(l <= 46.0) && (l <= 8e+91))) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * (J + (J * (K * (K * -0.125)))));
	} 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))
	tmp = 0
	if l <= -9.2e+132:
		tmp = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	elif (l <= -0.0245) or (not (l <= 46.0) and (l <= 8e+91)):
		tmp = U + ((math.exp(l) - math.exp(-l)) * (J + (J * (K * (K * -0.125)))))
	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))
	tmp = 0.0
	if (l <= -9.2e+132)
		tmp = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))));
	elseif ((l <= -0.0245) || (!(l <= 46.0) && (l <= 8e+91)))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * Float64(J + Float64(J * Float64(K * Float64(K * -0.125))))));
	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));
	tmp = 0.0;
	if (l <= -9.2e+132)
		tmp = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	elseif ((l <= -0.0245) || (~((l <= 46.0)) && (l <= 8e+91)))
		tmp = U + ((exp(l) - exp(-l)) * (J + (J * (K * (K * -0.125)))));
	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]}, If[LessEqual[l, -9.2e+132], N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -0.0245], And[N[Not[LessEqual[l, 46.0]], $MachinePrecision], LessEqual[l, 8e+91]]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * N[(J + N[(J * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if l < -9.2000000000000006e132

   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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if -9.2000000000000006e132 < l < -0.024500000000000001 or 46 < l < 8.00000000000000064e91

   1. Initial program 98.0%

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

   2. Taylor expanded in K around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*r* 0.0%

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

      5. distribute-rgt-out 83.4%

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

      6. *-commutative 83.4%

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

      7. *-commutative 83.4%

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

      8. unpow2 83.4%

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

      9. associate-*l* 83.4%

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

   4. Simplified 83.4%

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

   if -0.024500000000000001 < l < 46 or 8.00000000000000064e91 < l

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0 99.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0245 \lor \neg \left(\ell \leq 46\right) \land \ell \leq 8 \cdot 10^{+91}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.047:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.003:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -5.5e+102)
     t_1
     (if (<= l -0.047)
       t_2
       (if (<= l 0.003)
         (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 4.2e+94) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -5.5e+102) {
		tmp = t_1;
	} else if (l <= -0.047) {
		tmp = t_2;
	} else if (l <= 0.003) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.2e+94) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-5.5d+102)) then
        tmp = t_1
    else if (l <= (-0.047d0)) then
        tmp = t_2
    else if (l <= 0.003d0) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 4.2d+94) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -5.5e+102) {
		tmp = t_1;
	} else if (l <= -0.047) {
		tmp = t_2;
	} else if (l <= 0.003) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.2e+94) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -5.5e+102:
		tmp = t_1
	elif l <= -0.047:
		tmp = t_2
	elif l <= 0.003:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 4.2e+94:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -5.5e+102)
		tmp = t_1;
	elseif (l <= -0.047)
		tmp = t_2;
	elseif (l <= 0.003)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 4.2e+94)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -5.5e+102)
		tmp = t_1;
	elseif (l <= -0.047)
		tmp = t_2;
	elseif (l <= 0.003)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 4.2e+94)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.5e+102], t$95$1, If[LessEqual[l, -0.047], t$95$2, If[LessEqual[l, 0.003], 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], If[LessEqual[l, 4.2e+94], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.49999999999999981e102 or 4.19999999999999979e94 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if -5.49999999999999981e102 < l < -0.047 or 0.0030000000000000001 < l < 4.19999999999999979e94

   1. Initial program 97.9%

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

   2. Taylor expanded in K around 0 68.2%

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

   if -0.047 < l < 0.0030000000000000001

   1. Initial program 78.4%

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

   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 3 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.047:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.003:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 5: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00088:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.00042:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -5.5e+102)
     t_1
     (if (<= l -0.00088)
       t_2
       (if (<= l 0.00042)
         (+ U (* t_0 (* l (* J 2.0))))
         (if (<= l 3e+94) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -5.5e+102) {
		tmp = t_1;
	} else if (l <= -0.00088) {
		tmp = t_2;
	} else if (l <= 0.00042) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else if (l <= 3e+94) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-5.5d+102)) then
        tmp = t_1
    else if (l <= (-0.00088d0)) then
        tmp = t_2
    else if (l <= 0.00042d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else if (l <= 3d+94) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -5.5e+102) {
		tmp = t_1;
	} else if (l <= -0.00088) {
		tmp = t_2;
	} else if (l <= 0.00042) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else if (l <= 3e+94) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -5.5e+102:
		tmp = t_1
	elif l <= -0.00088:
		tmp = t_2
	elif l <= 0.00042:
		tmp = U + (t_0 * (l * (J * 2.0)))
	elif l <= 3e+94:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -5.5e+102)
		tmp = t_1;
	elseif (l <= -0.00088)
		tmp = t_2;
	elseif (l <= 0.00042)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	elseif (l <= 3e+94)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -5.5e+102)
		tmp = t_1;
	elseif (l <= -0.00088)
		tmp = t_2;
	elseif (l <= 0.00042)
		tmp = U + (t_0 * (l * (J * 2.0)));
	elseif (l <= 3e+94)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.5e+102], t$95$1, If[LessEqual[l, -0.00088], t$95$2, If[LessEqual[l, 0.00042], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e+94], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.49999999999999981e102 or 3.0000000000000001e94 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if -5.49999999999999981e102 < l < -8.80000000000000031e-4 or 4.2000000000000002e-4 < l < 3.0000000000000001e94

   1. Initial program 97.9%

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

   2. Taylor expanded in K around 0 68.2%

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

   if -8.80000000000000031e-4 < l < 4.2000000000000002e-4

   1. Initial program 78.4%

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

   2. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00088:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.00042:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+94}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 6: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.75:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.75)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ 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 tmp;
	if (t_0 <= 0.75) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} 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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.75d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    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 tmp;
	if (t_0 <= 0.75) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} 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))
	tmp = 0
	if t_0 <= 0.75:
		tmp = U + (t_0 * (l * (J * 2.0)))
	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))
	tmp = 0.0
	if (t_0 <= 0.75)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	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));
	tmp = 0.0;
	if (t_0 <= 0.75)
		tmp = U + (t_0 * (l * (J * 2.0)));
	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]}, If[LessEqual[t$95$0, 0.75], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.75

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

   4. Simplified 62.9%

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

   if 0.75 < (cos.f64 (/.f64 K 2))

   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 86.4%

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

   3. Taylor expanded in K around 0 84.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 70.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+91}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+161} \lor \neg \left(\ell \leq 1.35 \cdot 10^{+228}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
   (if (<= l -1.9e+28)
     t_0
     (if (<= l 800000.0)
       (+ U (* 2.0 (* l J)))
       (if (<= l 8.4e+91)
         (pow U -3.0)
         (if (or (<= l 1.95e+161) (not (<= l 1.35e+228)))
           t_0
           (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -1.9e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 8.4e+91) {
		tmp = pow(U, -3.0);
	} else if ((l <= 1.95e+161) || !(l <= 1.35e+228)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    if (l <= (-1.9d+28)) then
        tmp = t_0
    else if (l <= 800000.0d0) then
        tmp = u + (2.0d0 * (l * j))
    else if (l <= 8.4d+91) then
        tmp = u ** (-3.0d0)
    else if ((l <= 1.95d+161) .or. (.not. (l <= 1.35d+228))) then
        tmp = t_0
    else
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -1.9e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 8.4e+91) {
		tmp = Math.pow(U, -3.0);
	} else if ((l <= 1.95e+161) || !(l <= 1.35e+228)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	tmp = 0
	if l <= -1.9e+28:
		tmp = t_0
	elif l <= 800000.0:
		tmp = U + (2.0 * (l * J))
	elif l <= 8.4e+91:
		tmp = math.pow(U, -3.0)
	elif (l <= 1.95e+161) or not (l <= 1.35e+228):
		tmp = t_0
	else:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))
	tmp = 0.0
	if (l <= -1.9e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif (l <= 8.4e+91)
		tmp = U ^ -3.0;
	elseif ((l <= 1.95e+161) || !(l <= 1.35e+228))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	tmp = 0.0;
	if (l <= -1.9e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = U + (2.0 * (l * J));
	elseif (l <= 8.4e+91)
		tmp = U ^ -3.0;
	elseif ((l <= 1.95e+161) || ~((l <= 1.35e+228)))
		tmp = t_0;
	else
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+28], t$95$0, If[LessEqual[l, 800000.0], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.4e+91], N[Power[U, -3.0], $MachinePrecision], If[Or[LessEqual[l, 1.95e+161], N[Not[LessEqual[l, 1.35e+228]], $MachinePrecision]], t$95$0, N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8999999999999999e28 or 8.40000000000000031e91 < l < 1.9500000000000001e161 or 1.3500000000000001e228 < 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 90.7%

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

   3. Taylor expanded in K around 0 73.9%

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

   4. Taylor expanded in J around inf 73.9%

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

   if -1.8999999999999999e28 < l < 8e5

   1. Initial program 79.6%

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

   2. Taylor expanded in l around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*l* 92.8%

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

   4. Simplified 92.8%

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

   5. Taylor expanded in K around 0 81.9%

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

   if 8e5 < l < 8.40000000000000031e91

   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 32.6%

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

   if 1.9500000000000001e161 < l < 1.3500000000000001e228

   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 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

   4. Simplified 37.3%

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

   5. Taylor expanded in K around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. associate-*r* 30.5%

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

      3. distribute-rgt-out 66.2%

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

      4. *-commutative 66.2%

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

      5. unpow2 66.2%

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

   7. Simplified 66.2%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.25 + 2\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+28}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+91}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+161} \lor \neg \left(\ell \leq 1.35 \cdot 10^{+228}\right):\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 8: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+93}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+160} \lor \neg \left(\ell \leq 3.5 \cdot 10^{+226}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
   (if (<= l -1.8e+28)
     t_0
     (if (<= l 800000.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 4.1e+93)
         (pow U -3.0)
         (if (or (<= l 2.45e+160) (not (<= l 3.5e+226)))
           t_0
           (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -1.8e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 4.1e+93) {
		tmp = pow(U, -3.0);
	} else if ((l <= 2.45e+160) || !(l <= 3.5e+226)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    if (l <= (-1.8d+28)) then
        tmp = t_0
    else if (l <= 800000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 4.1d+93) then
        tmp = u ** (-3.0d0)
    else if ((l <= 2.45d+160) .or. (.not. (l <= 3.5d+226))) then
        tmp = t_0
    else
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -1.8e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 4.1e+93) {
		tmp = Math.pow(U, -3.0);
	} else if ((l <= 2.45e+160) || !(l <= 3.5e+226)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	tmp = 0
	if l <= -1.8e+28:
		tmp = t_0
	elif l <= 800000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 4.1e+93:
		tmp = math.pow(U, -3.0)
	elif (l <= 2.45e+160) or not (l <= 3.5e+226):
		tmp = t_0
	else:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))
	tmp = 0.0
	if (l <= -1.8e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 4.1e+93)
		tmp = U ^ -3.0;
	elseif ((l <= 2.45e+160) || !(l <= 3.5e+226))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	tmp = 0.0;
	if (l <= -1.8e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 4.1e+93)
		tmp = U ^ -3.0;
	elseif ((l <= 2.45e+160) || ~((l <= 3.5e+226)))
		tmp = t_0;
	else
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.8e+28], t$95$0, If[LessEqual[l, 800000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.1e+93], N[Power[U, -3.0], $MachinePrecision], If[Or[LessEqual[l, 2.45e+160], N[Not[LessEqual[l, 3.5e+226]], $MachinePrecision]], t$95$0, N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8e28 or 4.1000000000000001e93 < l < 2.4500000000000001e160 or 3.4999999999999998e226 < 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 90.7%

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

   3. Taylor expanded in K around 0 73.9%

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

   4. Taylor expanded in J around inf 73.9%

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

   if -1.8e28 < l < 8e5

   1. Initial program 79.6%

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

   2. Taylor expanded in l around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. *-commutative 92.8%

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

      3. associate-*l* 92.7%

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

   4. Simplified 92.7%

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

   if 8e5 < l < 4.1000000000000001e93

   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 32.6%

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

   if 2.4500000000000001e160 < l < 3.4999999999999998e226

   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 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

   4. Simplified 37.3%

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

   5. Taylor expanded in K around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. associate-*r* 30.5%

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

      3. distribute-rgt-out 66.2%

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

      4. *-commutative 66.2%

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

      5. unpow2 66.2%

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

   7. Simplified 66.2%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.25 + 2\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+28}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+93}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+160} \lor \neg \left(\ell \leq 3.5 \cdot 10^{+226}\right):\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+92}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+161} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+226}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
   (if (<= l -4.2e+28)
     t_0
     (if (<= l 800000.0)
       (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
       (if (<= l 4e+92)
         (pow U -3.0)
         (if (or (<= l 1.95e+161) (not (<= l 3.4e+226)))
           t_0
           (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -4.2e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 4e+92) {
		tmp = pow(U, -3.0);
	} else if ((l <= 1.95e+161) || !(l <= 3.4e+226)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    if (l <= (-4.2d+28)) then
        tmp = t_0
    else if (l <= 800000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 4d+92) then
        tmp = u ** (-3.0d0)
    else if ((l <= 1.95d+161) .or. (.not. (l <= 3.4d+226))) then
        tmp = t_0
    else
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	double tmp;
	if (l <= -4.2e+28) {
		tmp = t_0;
	} else if (l <= 800000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 4e+92) {
		tmp = Math.pow(U, -3.0);
	} else if ((l <= 1.95e+161) || !(l <= 3.4e+226)) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	tmp = 0
	if l <= -4.2e+28:
		tmp = t_0
	elif l <= 800000.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 4e+92:
		tmp = math.pow(U, -3.0)
	elif (l <= 1.95e+161) or not (l <= 3.4e+226):
		tmp = t_0
	else:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))
	tmp = 0.0
	if (l <= -4.2e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 4e+92)
		tmp = U ^ -3.0;
	elseif ((l <= 1.95e+161) || !(l <= 3.4e+226))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	tmp = 0.0;
	if (l <= -4.2e+28)
		tmp = t_0;
	elseif (l <= 800000.0)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 4e+92)
		tmp = U ^ -3.0;
	elseif ((l <= 1.95e+161) || ~((l <= 3.4e+226)))
		tmp = t_0;
	else
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2e+28], t$95$0, If[LessEqual[l, 800000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+92], N[Power[U, -3.0], $MachinePrecision], If[Or[LessEqual[l, 1.95e+161], N[Not[LessEqual[l, 3.4e+226]], $MachinePrecision]], t$95$0, N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.19999999999999978e28 or 4.0000000000000002e92 < l < 1.9500000000000001e161 or 3.39999999999999979e226 < 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 90.7%

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

   3. Taylor expanded in K around 0 73.9%

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

   4. Taylor expanded in J around inf 73.9%

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

   if -4.19999999999999978e28 < l < 8e5

   1. Initial program 79.6%

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

   2. Taylor expanded in l around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*l* 92.8%

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

   4. Simplified 92.8%

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

   if 8e5 < l < 4.0000000000000002e92

   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 32.6%

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

   if 1.9500000000000001e161 < l < 3.39999999999999979e226

   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 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

   4. Simplified 37.3%

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

   5. Taylor expanded in K around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. associate-*r* 30.5%

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

      3. distribute-rgt-out 66.2%

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

      4. *-commutative 66.2%

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

      5. unpow2 66.2%

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

   7. Simplified 66.2%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.25 + 2\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+28}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 800000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+92}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+161} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+226}\right):\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 10: 59.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-21} \lor \neg \left(\ell \leq 550\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7e-21) (not (<= l 550.0)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7e-21) || !(l <= 550.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-7d-21)) .or. (.not. (l <= 550.0d0))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7e-21) || !(l <= 550.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7e-21) or not (l <= 550.0):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7e-21) || !(l <= 550.0))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7e-21) || ~((l <= 550.0)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7e-21], N[Not[LessEqual[l, 550.0]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.0000000000000007e-21 or 550 < l

   1. Initial program 97.3%

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

   2. Taylor expanded in l around 0 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*l* 29.0%

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

   4. Simplified 29.0%

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

   5. Taylor expanded in K around 0 23.0%

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

      1. +-commutative 23.0%

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

      2. associate-*r* 23.0%

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

      3. distribute-rgt-out 40.7%

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

      4. *-commutative 40.7%

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

      5. unpow2 40.7%

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

   7. Simplified 40.7%

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

   if -7.0000000000000007e-21 < l < 550

   1. Initial program 79.7%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in K around 0 87.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-21} \lor \neg \left(\ell \leq 550\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 11: 53.8% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in l around 0 65.2%

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

   1. *-commutative 65.2%

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

   2. associate-*l* 65.2%

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

4. Simplified 65.2%

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

5. Taylor expanded in K around 0 57.4%

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

6. Final simplification 57.4%

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

Alternative 12: 39.3% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -720.0) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -720.0) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-720.0d0)) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -720.0) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -720.0:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -720.0)
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -720.0)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -720.0], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -720

   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 17.6%

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

   if -720 < l

   1. Initial program 85.0%

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

   2. Taylor expanded in J around 0 53.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 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 88.2%

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

3. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

5. Simplified 2.6%

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

6. Final simplification 2.6%

   \[\leadsto 1 \]

Alternative 14: 36.4% 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 88.2%

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

2. Taylor expanded in J around 0 42.5%

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

3. Final simplification 42.5%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023217 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))