Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.8%

Time: 16.7s

Alternatives: 14

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, t_0 \cdot t_1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 (- INFINITY))
     (+ (* t_1 (* t_0 J)) U)
     (if (<= t_0 5e-7)
       (+ U (* t_1 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
       (fma J (* t_0 t_1) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (t_1 * (t_0 * J)) + U;
	} else if (t_0 <= 5e-7) {
		tmp = U + (t_1 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = fma(J, (t_0 * t_1), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(t_1 * Float64(t_0 * J)) + U);
	elseif (t_0 <= 5e-7)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	else
		tmp = fma(J, Float64(t_0 * t_1), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(t$95$1 * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 5e-7], N[(U + N[(t$95$1 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(t$95$0 * t$95$1), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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))) < 4.99999999999999977e-7

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 4.99999999999999977e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-7)))
     (+ (* t_0 (* t_1 J)) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-7)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e-7)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-7):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-7))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e-7)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-7]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 4.99999999999999977e-7 < (-.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))) < 4.99999999999999977e-7

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := U + t_1 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -350:\\ \;\;\;\;U + t_0 \cdot \left(J + -0.125 \cdot \left(K \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.21:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;U + t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l))))
        (t_1 (cos (/ K 2.0)))
        (t_2 (+ U (* t_1 (* (pow l 3.0) (* J 0.3333333333333333))))))
   (if (<= l -4.5e+76)
     t_2
     (if (<= l -350.0)
       (+ U (* t_0 (+ J (* -0.125 (* K (* J K))))))
       (if (<= l 0.21)
         (+ U (* t_1 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 4.5e+68) (+ U (* t_0 J)) t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double t_2 = U + (t_1 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double tmp;
	if (l <= -4.5e+76) {
		tmp = t_2;
	} else if (l <= -350.0) {
		tmp = U + (t_0 * (J + (-0.125 * (K * (J * K)))));
	} else if (l <= 0.21) {
		tmp = U + (t_1 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.5e+68) {
		tmp = U + (t_0 * J);
	} else {
		tmp = t_2;
	}
	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 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    t_2 = u + (t_1 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    if (l <= (-4.5d+76)) then
        tmp = t_2
    else if (l <= (-350.0d0)) then
        tmp = u + (t_0 * (j + ((-0.125d0) * (k * (j * k)))))
    else if (l <= 0.21d0) then
        tmp = u + (t_1 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 4.5d+68) then
        tmp = u + (t_0 * j)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = U + (t_1 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double tmp;
	if (l <= -4.5e+76) {
		tmp = t_2;
	} else if (l <= -350.0) {
		tmp = U + (t_0 * (J + (-0.125 * (K * (J * K)))));
	} else if (l <= 0.21) {
		tmp = U + (t_1 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.5e+68) {
		tmp = U + (t_0 * J);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	t_2 = U + (t_1 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	tmp = 0
	if l <= -4.5e+76:
		tmp = t_2
	elif l <= -350.0:
		tmp = U + (t_0 * (J + (-0.125 * (K * (J * K)))))
	elif l <= 0.21:
		tmp = U + (t_1 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 4.5e+68:
		tmp = U + (t_0 * J)
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(U + Float64(t_1 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	tmp = 0.0
	if (l <= -4.5e+76)
		tmp = t_2;
	elseif (l <= -350.0)
		tmp = Float64(U + Float64(t_0 * Float64(J + Float64(-0.125 * Float64(K * Float64(J * K))))));
	elseif (l <= 0.21)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 4.5e+68)
		tmp = Float64(U + Float64(t_0 * J));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	t_2 = U + (t_1 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	tmp = 0.0;
	if (l <= -4.5e+76)
		tmp = t_2;
	elseif (l <= -350.0)
		tmp = U + (t_0 * (J + (-0.125 * (K * (J * K)))));
	elseif (l <= 0.21)
		tmp = U + (t_1 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 4.5e+68)
		tmp = U + (t_0 * J);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(t$95$1 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+76], t$95$2, If[LessEqual[l, -350.0], N[(U + N[(t$95$0 * N[(J + N[(-0.125 * N[(K * N[(J * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.21], N[(U + N[(t$95$1 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+68], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.4999999999999997e76 or 4.5000000000000003e68 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 95.8%

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

   3. Taylor expanded in l around inf 95.8%

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

      1. associate-*r* 95.8%

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

      2. *-commutative 95.8%

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

      3. associate-*r* 95.8%

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

   5. Simplified 95.8%

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

   if -4.4999999999999997e76 < l < -350

   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 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*r* 0.0%

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

      5. distribute-rgt-out 85.7%

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

      6. associate-*l* 85.7%

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

      7. unpow2 85.7%

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

      8. associate-*l* 85.7%

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

   4. Simplified 85.7%

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

   if -350 < l < 0.209999999999999992

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 0.209999999999999992 < l < 4.5000000000000003e68

   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 74.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -350:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + -0.125 \cdot \left(K \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.21:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.0062:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -2.7e+99)
     t_0
     (if (<= l -340.0)
       t_1
       (if (<= l 0.0062)
         (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
         (if (<= l 4.5e+68) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -2.7e+99) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = t_1;
	} else if (l <= 0.0062) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 4.5e+68) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-2.7d+99)) then
        tmp = t_0
    else if (l <= (-340.0d0)) then
        tmp = t_1
    else if (l <= 0.0062d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 4.5d+68) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -2.7e+99) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = t_1;
	} else if (l <= 0.0062) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 4.5e+68) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -2.7e+99:
		tmp = t_0
	elif l <= -340.0:
		tmp = t_1
	elif l <= 0.0062:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 4.5e+68:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -2.7e+99)
		tmp = t_0;
	elseif (l <= -340.0)
		tmp = t_1;
	elseif (l <= 0.0062)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 4.5e+68)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -2.7e+99)
		tmp = t_0;
	elseif (l <= -340.0)
		tmp = t_1;
	elseif (l <= 0.0062)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 4.5e+68)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.7e+99], t$95$0, If[LessEqual[l, -340.0], t$95$1, If[LessEqual[l, 0.0062], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+68], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.69999999999999989e99 or 4.5000000000000003e68 < 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 96.6%

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

   3. Taylor expanded in l around inf 96.6%

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

      1. associate-*r* 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*r* 96.6%

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

   5. Simplified 96.6%

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

   if -2.69999999999999989e99 < l < -340 or 0.00619999999999999978 < l < 4.5000000000000003e68

   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 74.2%

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

   if -340 < l < 0.00619999999999999978

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+99}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.0062:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 5: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + K \cdot \left(K \cdot -0.041666666666666664\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+244}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.0033:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (* J (pow l 3.0))
           (+ 0.3333333333333333 (* K (* K -0.041666666666666664))))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -6e+244)
     t_0
     (if (<= l -340.0)
       t_1
       (if (<= l 0.0033)
         (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
         (if (<= l 2e+137) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * pow(l, 3.0)) * (0.3333333333333333 + (K * (K * -0.041666666666666664))));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -6e+244) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = t_1;
	} else if (l <= 0.0033) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 2e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((j * (l ** 3.0d0)) * (0.3333333333333333d0 + (k * (k * (-0.041666666666666664d0)))))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-6d+244)) then
        tmp = t_0
    else if (l <= (-340.0d0)) then
        tmp = t_1
    else if (l <= 0.0033d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 2d+137) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * Math.pow(l, 3.0)) * (0.3333333333333333 + (K * (K * -0.041666666666666664))));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -6e+244) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = t_1;
	} else if (l <= 0.0033) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 2e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * math.pow(l, 3.0)) * (0.3333333333333333 + (K * (K * -0.041666666666666664))))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -6e+244:
		tmp = t_0
	elif l <= -340.0:
		tmp = t_1
	elif l <= 0.0033:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 2e+137:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * (l ^ 3.0)) * Float64(0.3333333333333333 + Float64(K * Float64(K * -0.041666666666666664)))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -6e+244)
		tmp = t_0;
	elseif (l <= -340.0)
		tmp = t_1;
	elseif (l <= 0.0033)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 2e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l ^ 3.0)) * (0.3333333333333333 + (K * (K * -0.041666666666666664))));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -6e+244)
		tmp = t_0;
	elseif (l <= -340.0)
		tmp = t_1;
	elseif (l <= 0.0033)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 2e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 + N[(K * N[(K * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e+244], t$95$0, If[LessEqual[l, -340.0], t$95$1, If[LessEqual[l, 0.0033], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.9999999999999995e244 or 2.0000000000000001e137 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({K}^{2} \cdot \left({\ell}^{3} \cdot J\right)\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + -0.041666666666666664 \cdot \left({K}^{2} \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]

      2. unpow2 0.0%

         \[\leadsto \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + -0.041666666666666664 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 0.0%

         \[\leadsto \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + \color{blue}{\left(-0.041666666666666664 \cdot \left(K \cdot K\right)\right) \cdot \left({\ell}^{3} \cdot J\right)}\right) + U \]

      4. distribute-rgt-out 86.4%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot J\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)} + U \]

      5. associate-*r* 86.4%

         \[\leadsto \left({\ell}^{3} \cdot J\right) \cdot \left(0.3333333333333333 + \color{blue}{\left(-0.041666666666666664 \cdot K\right) \cdot K}\right) + U \]

   8. Simplified 86.4%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot J\right) \cdot \left(0.3333333333333333 + \left(-0.041666666666666664 \cdot K\right) \cdot K\right)} + U \]

   if -5.9999999999999995e244 < l < -340 or 0.0033 < l < 2.0000000000000001e137

   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 76.9%

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

   if -340 < l < 0.0033

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+244}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + K \cdot \left(K \cdot -0.041666666666666664\right)\right)\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.0033:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+137}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + K \cdot \left(K \cdot -0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 6: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (+ U (* 2.0 (* l (* J (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)) <= -0.002) {
		tmp = U + (2.0 * (l * (J * 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)) <= (-0.002d0)) then
        tmp = u + (2.0d0 * (l * (j * 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)) <= -0.002) {
		tmp = U + (2.0 * (l * (J * 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)) <= -0.002:
		tmp = U + (2.0 * (l * (J * 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)) <= -0.002)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * 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)) <= -0.002)
		tmp = U + (2.0 * (l * (J * 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], -0.002], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $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)) < -2e-3

   1. Initial program 86.4%

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

   2. Taylor expanded in l around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. *-commutative 69.0%

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

      3. associate-*l* 69.0%

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

   4. Simplified 69.0%

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

   if -2e-3 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in K around 0 85.6%

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

   3. Taylor expanded in l around 0 86.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (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 + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
end

Wolfram

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

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in l around 0 89.6%

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

3. Final simplification 89.6%

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

Alternative 8: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{+39}:\\ \;\;\;\;U + \left|K \cdot \left(J \cdot K\right)\right|\\ \mathbf{elif}\;\ell \leq 620:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))
   (if (<= l -9.6e+242)
     t_0
     (if (<= l -1.02e+39)
       (+ U (fabs (* K (* J K))))
       (if (<= l 620.0) t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * (J * cos((K * 0.5)))));
	double tmp;
	if (l <= -9.6e+242) {
		tmp = t_0;
	} else if (l <= -1.02e+39) {
		tmp = U + fabs((K * (J * K)));
	} else if (l <= 620.0) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    if (l <= (-9.6d+242)) then
        tmp = t_0
    else if (l <= (-1.02d+39)) then
        tmp = u + abs((k * (j * k)))
    else if (l <= 620.0d0) then
        tmp = t_0
    else
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	double tmp;
	if (l <= -9.6e+242) {
		tmp = t_0;
	} else if (l <= -1.02e+39) {
		tmp = U + Math.abs((K * (J * K)));
	} else if (l <= 620.0) {
		tmp = t_0;
	} else {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	tmp = 0
	if l <= -9.6e+242:
		tmp = t_0
	elif l <= -1.02e+39:
		tmp = U + math.fabs((K * (J * K)))
	elif l <= 620.0:
		tmp = t_0
	else:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))))
	tmp = 0.0
	if (l <= -9.6e+242)
		tmp = t_0;
	elseif (l <= -1.02e+39)
		tmp = Float64(U + abs(Float64(K * Float64(J * K))));
	elseif (l <= 620.0)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (l * (J * cos((K * 0.5)))));
	tmp = 0.0;
	if (l <= -9.6e+242)
		tmp = t_0;
	elseif (l <= -1.02e+39)
		tmp = U + abs((K * (J * K)));
	elseif (l <= 620.0)
		tmp = t_0;
	else
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.6e+242], t$95$0, If[LessEqual[l, -1.02e+39], N[(U + N[Abs[N[(K * N[(J * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 620.0], t$95$0, N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.60000000000000049e242 or -1.02e39 < l < 620

   1. Initial program 77.6%

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

   2. Taylor expanded in l around 0 92.7%

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

      1. associate-*r* 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*l* 92.8%

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

   4. Simplified 92.8%

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

   if -9.60000000000000049e242 < l < -1.02e39

   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 3.6%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 16.1%

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

      1. distribute-rgt-out 16.1%

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

      2. unpow2 16.1%

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

   6. Simplified 16.1%

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

   7. Taylor expanded in K around inf 16.0%

      \[\leadsto \color{blue}{{K}^{2} \cdot J} + U \]
   8. Step-by-step derivation

      1. unpow2 16.0%

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

      2. *-commutative 16.0%

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

   9. Simplified 16.0%

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

       1. add-sqr-sqrt 13.2%

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

       2. sqrt-unprod 34.8%

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

       3. swap-sqr 27.2%

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

       4. pow2 27.2%

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

       5. pow2 27.2%

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

       6. pow-prod-up 27.2%

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

       7. metadata-eval 27.2%

          \[\leadsto \sqrt{\left(J \cdot J\right) \cdot {K}^{\color{blue}{4}}} + U \]

   11. Applied egg-rr 27.2%

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

       1. metadata-eval 27.2%

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

       2. pow-sqr 27.2%

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

       3. unswap-sqr 34.8%

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

       4. unpow2 34.8%

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

       5. associate-*r* 34.8%

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

       6. *-commutative 34.8%

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

       7. unpow2 34.8%

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

       8. associate-*r* 32.4%

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

       9. *-commutative 32.4%

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

       10. rem-sqrt-square 32.5%

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

       11. *-commutative 32.5%

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

   13. Simplified 32.5%

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

   if 620 < 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 22.9%

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

      1. associate-*r* 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*l* 22.9%

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

   4. Simplified 22.9%

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

   5. Taylor expanded in K around 0 16.5%

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

      1. +-commutative 16.5%

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

      2. associate-*r* 16.5%

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

      3. distribute-rgt-out 34.1%

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

      4. *-commutative 34.1%

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

      5. unpow2 34.1%

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

   7. Simplified 34.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+242}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{+39}:\\ \;\;\;\;U + \left|K \cdot \left(J \cdot K\right)\right|\\ \mathbf{elif}\;\ell \leq 620:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 9: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+67}:\\ \;\;\;\;U + \left|K \cdot \left(J \cdot K\right)\right|\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+34} \lor \neg \left(\ell \leq 360\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -7.5e+242)
     t_0
     (if (<= l -9e+67)
       (+ U (fabs (* K (* J K))))
       (if (or (<= l -3.2e+34) (not (<= l 360.0)))
         t_0
         (+ U (* 2.0 (* l J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -7.5e+242) {
		tmp = t_0;
	} else if (l <= -9e+67) {
		tmp = U + fabs((K * (J * K)));
	} else if ((l <= -3.2e+34) || !(l <= 360.0)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (l <= (-7.5d+242)) then
        tmp = t_0
    else if (l <= (-9d+67)) then
        tmp = u + abs((k * (j * k)))
    else if ((l <= (-3.2d+34)) .or. (.not. (l <= 360.0d0))) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -7.5e+242) {
		tmp = t_0;
	} else if (l <= -9e+67) {
		tmp = U + Math.abs((K * (J * K)));
	} else if ((l <= -3.2e+34) || !(l <= 360.0)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if l <= -7.5e+242:
		tmp = t_0
	elif l <= -9e+67:
		tmp = U + math.fabs((K * (J * K)))
	elif (l <= -3.2e+34) or not (l <= 360.0):
		tmp = t_0
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -7.5e+242)
		tmp = t_0;
	elseif (l <= -9e+67)
		tmp = Float64(U + abs(Float64(K * Float64(J * K))));
	elseif ((l <= -3.2e+34) || !(l <= 360.0))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (l <= -7.5e+242)
		tmp = t_0;
	elseif (l <= -9e+67)
		tmp = U + abs((K * (J * K)));
	elseif ((l <= -3.2e+34) || ~((l <= 360.0)))
		tmp = t_0;
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.5e+242], t$95$0, If[LessEqual[l, -9e+67], N[(U + N[Abs[N[(K * N[(J * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -3.2e+34], N[Not[LessEqual[l, 360.0]], $MachinePrecision]], t$95$0, N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.49999999999999961e242 or -8.9999999999999997e67 < l < -3.1999999999999998e34 or 360 < 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 29.4%

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

      1. associate-*r* 29.4%

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

      2. *-commutative 29.4%

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

      3. associate-*l* 29.4%

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

   4. Simplified 29.4%

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

   5. Taylor expanded in K around 0 17.3%

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

      1. +-commutative 17.3%

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

      2. associate-*r* 17.3%

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

      3. distribute-rgt-out 40.7%

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

      4. *-commutative 40.7%

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

      5. unpow2 40.7%

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

   7. Simplified 40.7%

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

   if -7.49999999999999961e242 < l < -8.9999999999999997e67

   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 3.5%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 13.0%

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

      1. distribute-rgt-out 13.0%

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

      2. unpow2 13.0%

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

   6. Simplified 13.0%

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

   7. Taylor expanded in K around inf 12.9%

      \[\leadsto \color{blue}{{K}^{2} \cdot J} + U \]
   8. Step-by-step derivation

      1. unpow2 12.9%

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

      2. *-commutative 12.9%

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

   9. Simplified 12.9%

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

       1. add-sqr-sqrt 9.7%

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

       2. sqrt-unprod 35.6%

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

       3. swap-sqr 26.7%

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

       4. pow2 26.7%

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

       5. pow2 26.7%

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

       6. pow-prod-up 26.7%

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

       7. metadata-eval 26.7%

          \[\leadsto \sqrt{\left(J \cdot J\right) \cdot {K}^{\color{blue}{4}}} + U \]

   11. Applied egg-rr 26.7%

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

       1. metadata-eval 26.7%

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

       2. pow-sqr 26.7%

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

       3. unswap-sqr 35.6%

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

       4. unpow2 35.6%

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

       5. associate-*r* 35.6%

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

       6. *-commutative 35.6%

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

       7. unpow2 35.6%

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

       8. associate-*r* 32.8%

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

       9. *-commutative 32.8%

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

       10. rem-sqrt-square 32.8%

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

       11. *-commutative 32.8%

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

   13. Simplified 32.8%

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

   if -3.1999999999999998e34 < l < 360

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in K around 0 85.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+242}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+67}:\\ \;\;\;\;U + \left|K \cdot \left(J \cdot K\right)\right|\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+34} \lor \neg \left(\ell \leq 360\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 10: 59.1% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+34} \lor \neg \left(\ell \leq 255\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.3e+34) (not (<= l 255.0)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.3e+34) || !(l <= 255.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.3d+34)) .or. (.not. (l <= 255.0d0))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.3e+34) || !(l <= 255.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.3e+34) or not (l <= 255.0):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.3e+34) || !(l <= 255.0))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.3e+34) || ~((l <= 255.0)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.3e+34], N[Not[LessEqual[l, 255.0]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.2999999999999998e34 or 255 < 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 25.9%

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

      1. associate-*r* 25.9%

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

      2. *-commutative 25.9%

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

      3. associate-*l* 25.9%

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

   4. Simplified 25.9%

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

   5. Taylor expanded in K around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. associate-*r* 14.1%

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

      3. distribute-rgt-out 34.0%

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

      4. *-commutative 34.0%

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

      5. unpow2 34.0%

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

   7. Simplified 34.0%

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

   if -2.2999999999999998e34 < l < 255

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in K around 0 85.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+34} \lor \neg \left(\ell \leq 255\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 11: 53.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in l around 0 64.9%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in K around 0 54.1%

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

6. Final simplification 54.1%

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

Alternative 12: 40.1% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 280000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 280000.0) U (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 280000.0) {
		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 <= 280000.0d0) 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 <= 280000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 280000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 280000.0)
		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 <= 280000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 280000.0], U, N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8e5

   1. Initial program 82.4%

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

      1. associate-*l* 82.4%

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

      2. fma-def 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in J around 0 49.8%

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

   if 2.8e5 < 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 15.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 280000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 13: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate-*l* 86.2%

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

   2. fma-def 86.2%

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

3. Simplified 86.2%

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

5. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto 1 \]

Alternative 14: 37.0% 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 86.2%

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

   1. associate-*l* 86.2%

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

   2. fma-def 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in J around 0 39.5%

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

5. Final simplification 39.5%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023279 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))