Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.8%

Time: 21.9s

Alternatives: 17

Speedup: 1.5×

• 49.3% 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: 86.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.02\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 0.02)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 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 <= 0.02)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((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 <= 0.02)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((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 <= 0.02):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((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 <= 0.02))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + 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 <= 0.02)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((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, 0.02]], $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[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $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 0.0200000000000000004 < (-.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))) < 0.0200000000000000004

   1. Initial program 66.2%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.02\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-9)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* (* l J) (* 2.0 (cos (* K 0.5))))))))

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 <= 1e-9)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	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 <= 1e-9)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	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 <= 1e-9):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	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 <= 1e-9))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	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 <= 1e-9)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	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, 1e-9]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $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 1.00000000000000006e-9 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000006e-9

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot t_0\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.000145:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* 0.016666666666666666 (* J t_0)))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -4.2e+90)
     t_1
     (if (<= l -8.5e-5)
       t_2
       (if (<= l 0.000145)
         (+ U (* (* l J) (* 2.0 t_0)))
         (if (<= l 4.2e+61) 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 + (pow(l, 5.0) * (0.016666666666666666 * (J * t_0)));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -4.2e+90) {
		tmp = t_1;
	} else if (l <= -8.5e-5) {
		tmp = t_2;
	} else if (l <= 0.000145) {
		tmp = U + ((l * J) * (2.0 * t_0));
	} else if (l <= 4.2e+61) {
		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 + ((l ** 5.0d0) * (0.016666666666666666d0 * (j * t_0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-4.2d+90)) then
        tmp = t_1
    else if (l <= (-8.5d-5)) then
        tmp = t_2
    else if (l <= 0.000145d0) then
        tmp = u + ((l * j) * (2.0d0 * t_0))
    else if (l <= 4.2d+61) 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 + (Math.pow(l, 5.0) * (0.016666666666666666 * (J * t_0)));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -4.2e+90) {
		tmp = t_1;
	} else if (l <= -8.5e-5) {
		tmp = t_2;
	} else if (l <= 0.000145) {
		tmp = U + ((l * J) * (2.0 * t_0));
	} else if (l <= 4.2e+61) {
		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 + (math.pow(l, 5.0) * (0.016666666666666666 * (J * t_0)))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -4.2e+90:
		tmp = t_1
	elif l <= -8.5e-5:
		tmp = t_2
	elif l <= 0.000145:
		tmp = U + ((l * J) * (2.0 * t_0))
	elif l <= 4.2e+61:
		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((l ^ 5.0) * Float64(0.016666666666666666 * Float64(J * t_0))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -4.2e+90)
		tmp = t_1;
	elseif (l <= -8.5e-5)
		tmp = t_2;
	elseif (l <= 0.000145)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * t_0)));
	elseif (l <= 4.2e+61)
		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 + ((l ^ 5.0) * (0.016666666666666666 * (J * t_0)));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -4.2e+90)
		tmp = t_1;
	elseif (l <= -8.5e-5)
		tmp = t_2;
	elseif (l <= 0.000145)
		tmp = U + ((l * J) * (2.0 * t_0));
	elseif (l <= 4.2e+61)
		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[(N[Power[l, 5.0], $MachinePrecision] * N[(0.016666666666666666 * N[(J * t$95$0), $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, -4.2e+90], t$95$1, If[LessEqual[l, -8.5e-5], t$95$2, If[LessEqual[l, 0.000145], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+61], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.19999999999999961e90 or 4.2000000000000002e61 < 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%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto 0.016666666666666666 \cdot \left(\color{blue}{\left({\ell}^{5} \cdot J\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

      3. *-commutative 100.0%

         \[\leadsto 0.016666666666666666 \cdot \left(\left({\ell}^{5} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      4. associate-*l* 100.0%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left({\ell}^{5} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5}\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot 0.016666666666666666\right)} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      7. associate-*l* 100.0%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 100.0%

         \[\leadsto {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -4.19999999999999961e90 < l < -8.500000000000001e-5 or 1.45e-4 < l < 4.2000000000000002e61

   1. Initial program 99.7%

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

   2. Taylor expanded in K around 0 87.2%

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

   if -8.500000000000001e-5 < l < 1.45e-4

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.000145:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot t_0\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -230:\\ \;\;\;\;U + t_2 \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq 0.00055:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_2 + U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* 0.016666666666666666 (* J t_0)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -4.4e+92)
     t_1
     (if (<= l -230.0)
       (+ U (* t_2 (+ 1.0 (* -0.125 (* K K)))))
       (if (<= l 0.00055)
         (+ U (* (* l J) (* 2.0 t_0)))
         (if (<= l 4.5e+61) (+ t_2 U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 5.0) * (0.016666666666666666 * (J * t_0)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -4.4e+92) {
		tmp = t_1;
	} else if (l <= -230.0) {
		tmp = U + (t_2 * (1.0 + (-0.125 * (K * K))));
	} else if (l <= 0.00055) {
		tmp = U + ((l * J) * (2.0 * t_0));
	} else if (l <= 4.5e+61) {
		tmp = t_2 + U;
	} 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 + ((l ** 5.0d0) * (0.016666666666666666d0 * (j * t_0)))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-4.4d+92)) then
        tmp = t_1
    else if (l <= (-230.0d0)) then
        tmp = u + (t_2 * (1.0d0 + ((-0.125d0) * (k * k))))
    else if (l <= 0.00055d0) then
        tmp = u + ((l * j) * (2.0d0 * t_0))
    else if (l <= 4.5d+61) then
        tmp = t_2 + u
    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 + (Math.pow(l, 5.0) * (0.016666666666666666 * (J * t_0)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -4.4e+92) {
		tmp = t_1;
	} else if (l <= -230.0) {
		tmp = U + (t_2 * (1.0 + (-0.125 * (K * K))));
	} else if (l <= 0.00055) {
		tmp = U + ((l * J) * (2.0 * t_0));
	} else if (l <= 4.5e+61) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 5.0) * (0.016666666666666666 * (J * t_0)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -4.4e+92:
		tmp = t_1
	elif l <= -230.0:
		tmp = U + (t_2 * (1.0 + (-0.125 * (K * K))))
	elif l <= 0.00055:
		tmp = U + ((l * J) * (2.0 * t_0))
	elif l <= 4.5e+61:
		tmp = t_2 + U
	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((l ^ 5.0) * Float64(0.016666666666666666 * Float64(J * t_0))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -4.4e+92)
		tmp = t_1;
	elseif (l <= -230.0)
		tmp = Float64(U + Float64(t_2 * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	elseif (l <= 0.00055)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * t_0)));
	elseif (l <= 4.5e+61)
		tmp = Float64(t_2 + U);
	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 + ((l ^ 5.0) * (0.016666666666666666 * (J * t_0)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -4.4e+92)
		tmp = t_1;
	elseif (l <= -230.0)
		tmp = U + (t_2 * (1.0 + (-0.125 * (K * K))));
	elseif (l <= 0.00055)
		tmp = U + ((l * J) * (2.0 * t_0));
	elseif (l <= 4.5e+61)
		tmp = t_2 + U;
	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[(N[Power[l, 5.0], $MachinePrecision] * N[(0.016666666666666666 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -4.4e+92], t$95$1, If[LessEqual[l, -230.0], N[(U + N[(t$95$2 * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00055], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+61], N[(t$95$2 + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.39999999999999984e92 or 4.5e61 < 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%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto 0.016666666666666666 \cdot \left(\color{blue}{\left({\ell}^{5} \cdot J\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

      3. *-commutative 100.0%

         \[\leadsto 0.016666666666666666 \cdot \left(\left({\ell}^{5} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      4. associate-*l* 100.0%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left({\ell}^{5} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5}\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot 0.016666666666666666\right)} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      7. associate-*l* 100.0%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 100.0%

         \[\leadsto {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -4.39999999999999984e92 < l < -230

   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 89.5%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   3. Step-by-step derivation

      1. unpow2 89.5%

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   4. Simplified 89.5%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} + U \]

   if -230 < l < 5.50000000000000033e-4

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 5.50000000000000033e-4 < l < 4.5e61

   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 84.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -230:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq 0.00055:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.08:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.08)
   (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.08) {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	} 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) :: tmp
    if (cos((k / 2.0d0)) <= 0.08d0) then
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    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 tmp;
	if (Math.cos((K / 2.0)) <= 0.08) {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.08:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.08)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.08)
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.08], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $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)) < 0.0800000000000000017

   1. Initial program 79.0%

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

   2. Taylor expanded in l around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*r* 74.6%

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

      3. associate-*l* 74.6%

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

      4. *-commutative 74.6%

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

   4. Simplified 74.6%

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

   if 0.0800000000000000017 < (cos.f64 (/.f64 K 2))

   1. Initial program 83.4%

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

   2. Taylor expanded in l around 0 88.2%

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

   3. Taylor expanded in K around 0 84.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.08:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 6: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0031 \lor \neg \left(\ell \leq 0.0007\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0031) (not (<= l 0.0007)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0031) || !(l <= 0.0007)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.0031d0)) .or. (.not. (l <= 0.0007d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0031) || !(l <= 0.0007)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.0031) or not (l <= 0.0007):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0031) || !(l <= 0.0007))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.0031) || ~((l <= 0.0007)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0031], N[Not[LessEqual[l, 0.0007]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.00309999999999999989 or 6.99999999999999993e-4 < 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 80.9%

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

   if -0.00309999999999999989 < l < 6.99999999999999993e-4

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0031 \lor \neg \left(\ell \leq 0.0007\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2200:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.3e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -2200.0)
     (fabs (* U (+ U 8.0)))
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -2200.0) {
		tmp = fabs((U * (U + 8.0)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-4.3d+71)) then
        tmp = u + ((l * 2.0d0) * (j * (1.0d0 + ((-0.125d0) * (k * k)))))
    else if (l <= (-2200.0d0)) then
        tmp = abs((u * (u + 8.0d0)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -2200.0) {
		tmp = Math.abs((U * (U + 8.0)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.3e+71:
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))))
	elif l <= -2200.0:
		tmp = math.fabs((U * (U + 8.0)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.3e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -2200.0)
		tmp = abs(Float64(U * Float64(U + 8.0)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.3e+71)
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	elseif (l <= -2200.0)
		tmp = abs((U * (U + 8.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.3e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2200.0], N[Abs[N[(U * N[(U + 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.29999999999999984e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -4.29999999999999984e71 < l < -2200

   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 43.5%

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

      1. add-sqr-sqrt 43.2%

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(U - -8\right)} \cdot \sqrt{U \cdot \left(U - -8\right)}} \]

      2. sqrt-unprod 43.5%

         \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(U - -8\right)\right) \cdot \left(U \cdot \left(U - -8\right)\right)}} \]

      3. pow2 43.5%

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

      4. sub-neg 43.5%

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

      5. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. rem-sqrt-square 43.6%

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

      3. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -2200 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2200:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 8: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1300:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.5e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -1300.0)
     (fabs (* U (+ U 8.0)))
     (+ U (* (* l 2.0) (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.5e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -1300.0) {
		tmp = fabs((U * (U + 8.0)));
	} else {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-3.5d+71)) then
        tmp = u + ((l * 2.0d0) * (j * (1.0d0 + ((-0.125d0) * (k * k)))))
    else if (l <= (-1300.0d0)) then
        tmp = abs((u * (u + 8.0d0)))
    else
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.5e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -1300.0) {
		tmp = Math.abs((U * (U + 8.0)));
	} else {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.5e+71:
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))))
	elif l <= -1300.0:
		tmp = math.fabs((U * (U + 8.0)))
	else:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.5e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -1300.0)
		tmp = abs(Float64(U * Float64(U + 8.0)));
	else
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.5e+71)
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	elseif (l <= -1300.0)
		tmp = abs((U * (U + 8.0)));
	else
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.5e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1300.0], N[Abs[N[(U * N[(U + 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999999e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -3.4999999999999999e71 < l < -1300

   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 43.5%

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

      1. add-sqr-sqrt 43.2%

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(U - -8\right)} \cdot \sqrt{U \cdot \left(U - -8\right)}} \]

      2. sqrt-unprod 43.5%

         \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(U - -8\right)\right) \cdot \left(U \cdot \left(U - -8\right)\right)}} \]

      3. pow2 43.5%

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

      4. sub-neg 43.5%

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

      5. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. rem-sqrt-square 43.6%

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

      3. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -1300 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*r* 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1300:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -900:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.5e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -900.0)
     (fabs (* U (+ U 8.0)))
     (+ U (* (* l J) (* 2.0 (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.5e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -900.0) {
		tmp = fabs((U * (U + 8.0)));
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-5.5d+71)) then
        tmp = u + ((l * 2.0d0) * (j * (1.0d0 + ((-0.125d0) * (k * k)))))
    else if (l <= (-900.0d0)) then
        tmp = abs((u * (u + 8.0d0)))
    else
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.5e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -900.0) {
		tmp = Math.abs((U * (U + 8.0)));
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.5e+71:
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))))
	elif l <= -900.0:
		tmp = math.fabs((U * (U + 8.0)))
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.5e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -900.0)
		tmp = abs(Float64(U * Float64(U + 8.0)));
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.5e+71)
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	elseif (l <= -900.0)
		tmp = abs((U * (U + 8.0)));
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.5e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -900.0], N[Abs[N[(U * N[(U + 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -5.5e71 < l < -900

   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 43.5%

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

      1. add-sqr-sqrt 43.2%

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(U - -8\right)} \cdot \sqrt{U \cdot \left(U - -8\right)}} \]

      2. sqrt-unprod 43.5%

         \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(U - -8\right)\right) \cdot \left(U \cdot \left(U - -8\right)\right)}} \]

      3. pow2 43.5%

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

      4. sub-neg 43.5%

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

      5. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. rem-sqrt-square 43.6%

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

      3. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -900 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*r* 77.0%

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

      3. associate-*l* 77.0%

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

      4. *-commutative 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -900:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.3e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -820.0) (fabs (* U (+ U 8.0))) (+ U (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -820.0) {
		tmp = fabs((U * (U + 8.0)));
	} 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 <= (-4.3d+71)) then
        tmp = u + ((l * 2.0d0) * (j * (1.0d0 + ((-0.125d0) * (k * k)))))
    else if (l <= (-820.0d0)) then
        tmp = abs((u * (u + 8.0d0)))
    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 <= -4.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -820.0) {
		tmp = Math.abs((U * (U + 8.0)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.3e+71:
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))))
	elif l <= -820.0:
		tmp = math.fabs((U * (U + 8.0)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.3e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -820.0)
		tmp = abs(Float64(U * Float64(U + 8.0)));
	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 <= -4.3e+71)
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	elseif (l <= -820.0)
		tmp = abs((U * (U + 8.0)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.3e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -820.0], N[Abs[N[(U * N[(U + 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.29999999999999984e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -4.29999999999999984e71 < l < -820

   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 43.5%

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

      1. add-sqr-sqrt 43.2%

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(U - -8\right)} \cdot \sqrt{U \cdot \left(U - -8\right)}} \]

      2. sqrt-unprod 43.5%

         \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(U - -8\right)\right) \cdot \left(U \cdot \left(U - -8\right)\right)}} \]

      3. pow2 43.5%

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

      4. sub-neg 43.5%

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

      5. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. rem-sqrt-square 43.6%

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

      3. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -820 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*r* 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in K around 0 64.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 11: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -720:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.7e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -720.0) (fabs (* U (+ U 8.0))) (fma (* l 2.0) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.7e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -720.0) {
		tmp = fabs((U * (U + 8.0)));
	} else {
		tmp = fma((l * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.7e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -720.0)
		tmp = abs(Float64(U * Float64(U + 8.0)));
	else
		tmp = fma(Float64(l * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.7e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -720.0], N[Abs[N[(U * N[(U + 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -3.7e71 < 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 43.5%

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

      1. add-sqr-sqrt 43.2%

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(U - -8\right)} \cdot \sqrt{U \cdot \left(U - -8\right)}} \]

      2. sqrt-unprod 43.5%

         \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(U - -8\right)\right) \cdot \left(U \cdot \left(U - -8\right)\right)}} \]

      3. pow2 43.5%

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

      4. sub-neg 43.5%

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

      5. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. rem-sqrt-square 43.6%

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

      3. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -720 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*r* 77.0%

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

   4. Simplified 77.0%

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

      1. fma-def 77.0%

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

      2. *-commutative 77.0%

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

      3. *-commutative 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in K around 0 64.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -720:\\ \;\;\;\;\left|U \cdot \left(U + 8\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \]

Alternative 12: 56.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -980:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.3e+71)
   (+ U (* (* l 2.0) (* J (+ 1.0 (* -0.125 (* K K))))))
   (if (<= l -980.0) (* U U) (+ U (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -980.0) {
		tmp = U * U;
	} 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 <= (-3.3d+71)) then
        tmp = u + ((l * 2.0d0) * (j * (1.0d0 + ((-0.125d0) * (k * k)))))
    else if (l <= (-980.0d0)) then
        tmp = u * u
    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 <= -3.3e+71) {
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	} else if (l <= -980.0) {
		tmp = U * U;
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.3e+71:
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))))
	elif l <= -980.0:
		tmp = U * U
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+71)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	elseif (l <= -980.0)
		tmp = Float64(U * U);
	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 <= -3.3e+71)
		tmp = U + ((l * 2.0) * (J * (1.0 + (-0.125 * (K * K)))));
	elseif (l <= -980.0)
		tmp = U * U;
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+71], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -980.0], N[(U * U), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2999999999999998e71

   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 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 29.9%

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

   4. Simplified 29.9%

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

   5. Taylor expanded in K around 0 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 39.3%

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   7. Simplified 39.3%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(J \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -3.2999999999999998e71 < l < -980

   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 43.5%

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

   if -980 < l

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*r* 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in K around 0 64.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+71}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -980:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 13: 42.4% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -850:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.45:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -850.0) (* U U) (if (<= l 1.45) U (* U (- U -8.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -850.0) {
		tmp = U * U;
	} else if (l <= 1.45) {
		tmp = U;
	} else {
		tmp = U * (U - -8.0);
	}
	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 <= (-850.0d0)) then
        tmp = u * u
    else if (l <= 1.45d0) then
        tmp = u
    else
        tmp = u * (u - (-8.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -850.0) {
		tmp = U * U;
	} else if (l <= 1.45) {
		tmp = U;
	} else {
		tmp = U * (U - -8.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -850.0:
		tmp = U * U
	elif l <= 1.45:
		tmp = U
	else:
		tmp = U * (U - -8.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -850.0)
		tmp = Float64(U * U);
	elseif (l <= 1.45)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -850.0)
		tmp = U * U;
	elseif (l <= 1.45)
		tmp = U;
	else
		tmp = U * (U - -8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -850.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.45], U, N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -850

   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 23.9%

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

   if -850 < l < 1.44999999999999996

   1. Initial program 66.7%

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

   2. Taylor expanded in J around 0 64.7%

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

   if 1.44999999999999996 < l

   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 10.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -850:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.45:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \end{array} \]

Alternative 14: 42.2% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+51}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-8 - U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -720.0) (* U U) (if (<= l 8.4e+51) U (- -8.0 (* U U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -720.0) {
		tmp = U * U;
	} else if (l <= 8.4e+51) {
		tmp = U;
	} else {
		tmp = -8.0 - (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 <= (-720.0d0)) then
        tmp = u * u
    else if (l <= 8.4d+51) then
        tmp = u
    else
        tmp = (-8.0d0) - (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 <= -720.0) {
		tmp = U * U;
	} else if (l <= 8.4e+51) {
		tmp = U;
	} else {
		tmp = -8.0 - (U * U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -720.0:
		tmp = U * U
	elif l <= 8.4e+51:
		tmp = U
	else:
		tmp = -8.0 - (U * U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -720.0)
		tmp = Float64(U * U);
	elseif (l <= 8.4e+51)
		tmp = U;
	else
		tmp = Float64(-8.0 - Float64(U * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -720.0)
		tmp = U * U;
	elseif (l <= 8.4e+51)
		tmp = U;
	else
		tmp = -8.0 - (U * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -720.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 8.4e+51], U, N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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 23.9%

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

   if -720 < l < 8.4000000000000005e51

   1. Initial program 68.5%

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

   2. Taylor expanded in J around 0 61.2%

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

   if 8.4000000000000005e51 < 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. 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 15.4%

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

      1. cancel-sign-sub-inv 15.4%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 15.4%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+51}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-8 - U \cdot U\\ \end{array} \]

Alternative 15: 41.9% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -720.0) (* U U) (if (<= l 1.8e+66) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -720.0) {
		tmp = U * U;
	} else if (l <= 1.8e+66) {
		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 <= (-720.0d0)) then
        tmp = u * u
    else if (l <= 1.8d+66) 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 <= -720.0) {
		tmp = U * U;
	} else if (l <= 1.8e+66) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -720.0:
		tmp = U * U
	elif l <= 1.8e+66:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -720.0)
		tmp = Float64(U * U);
	elseif (l <= 1.8e+66)
		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 <= -720.0)
		tmp = U * U;
	elseif (l <= 1.8e+66)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -720.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.8e+66], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -720 or 1.8e66 < l

   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 18.6%

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

   if -720 < l < 1.8e66

   1. Initial program 68.9%

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

   2. Taylor expanded in J around 0 60.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 16: 54.2% 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 82.4%

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

2. Taylor expanded in l around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. associate-*l* 64.6%

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

   3. *-commutative 64.6%

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

   4. associate-*r* 64.6%

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

4. Simplified 64.6%

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

5. Taylor expanded in K around 0 53.9%

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

6. Final simplification 53.9%

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

Alternative 17: 36.9% 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 82.4%

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

2. Taylor expanded in J around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023299 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))