Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.4%

Time: 9.1s

Alternatives: 12

Speedup: 1.4×

• 48.9% 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: 86.1% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-15)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* 2.0 (* J (* l (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 <= 2e-15)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * 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 <= 2e-15)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * 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 <= 2e-15):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (2.0 * (J * (l * 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 <= 2e-15))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	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)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e-15)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (2.0 * (J * (l * 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, 2e-15]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $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 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000002e-15 < (-.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))) < 2.0000000000000002e-15

   1. Initial program 64.5%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 2: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-11}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -2.6e+73)
     t_1
     (if (<= l -3.8e+25)
       t_2
       (if (<= l 8e-11)
         (+ U (* t_0 (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0)))))
         (if (<= l 7.8e+91) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.6e+73) {
		tmp = t_1;
	} else if (l <= -3.8e+25) {
		tmp = t_2;
	} else if (l <= 8e-11) {
		tmp = U + (t_0 * (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	} else if (l <= 7.8e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.6d+73)) then
        tmp = t_1
    else if (l <= (-3.8d+25)) then
        tmp = t_2
    else if (l <= 8d-11) then
        tmp = u + (t_0 * (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (l * 2.0d0))))
    else if (l <= 7.8d+91) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.6e+73) {
		tmp = t_1;
	} else if (l <= -3.8e+25) {
		tmp = t_2;
	} else if (l <= 8e-11) {
		tmp = U + (t_0 * (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	} else if (l <= 7.8e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -2.6e+73:
		tmp = t_1
	elif l <= -3.8e+25:
		tmp = t_2
	elif l <= 8e-11:
		tmp = U + (t_0 * (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))))
	elif l <= 7.8e+91:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.6e+73)
		tmp = t_1;
	elseif (l <= -3.8e+25)
		tmp = t_2;
	elseif (l <= 8e-11)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0)))));
	elseif (l <= 7.8e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -2.6e+73)
		tmp = t_1;
	elseif (l <= -3.8e+25)
		tmp = t_2;
	elseif (l <= 8e-11)
		tmp = U + (t_0 * (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0))));
	elseif (l <= 7.8e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.6e+73], t$95$1, If[LessEqual[l, -3.8e+25], t$95$2, If[LessEqual[l, 8e-11], N[(U + N[(t$95$0 * N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+91], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6000000000000001e73 or 7.79999999999999935e91 < 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 98.1%

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

   3. Taylor expanded in l around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

   5. Simplified 98.1%

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

   if -2.6000000000000001e73 < l < -3.8e25 or 7.99999999999999952e-11 < l < 7.79999999999999935e91

   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%

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

   if -3.8e25 < l < 7.99999999999999952e-11

   1. Initial program 65.3%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-11}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+91}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -2.6e+73)
     t_0
     (if (<= l -3.8e+25)
       t_1
       (if (<= l 8e-11)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (if (<= l 4.9e+98) 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 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.6e+73) {
		tmp = t_0;
	} else if (l <= -3.8e+25) {
		tmp = t_1;
	} else if (l <= 8e-11) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 4.9e+98) {
		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 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.6d+73)) then
        tmp = t_0
    else if (l <= (-3.8d+25)) then
        tmp = t_1
    else if (l <= 8d-11) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 4.9d+98) 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 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.6e+73) {
		tmp = t_0;
	} else if (l <= -3.8e+25) {
		tmp = t_1;
	} else if (l <= 8e-11) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 4.9e+98) {
		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 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -2.6e+73:
		tmp = t_0
	elif l <= -3.8e+25:
		tmp = t_1
	elif l <= 8e-11:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 4.9e+98:
		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(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.6e+73)
		tmp = t_0;
	elseif (l <= -3.8e+25)
		tmp = t_1;
	elseif (l <= 8e-11)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 4.9e+98)
		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 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -2.6e+73)
		tmp = t_0;
	elseif (l <= -3.8e+25)
		tmp = t_1;
	elseif (l <= 8e-11)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 4.9e+98)
		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[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.6e+73], t$95$0, If[LessEqual[l, -3.8e+25], t$95$1, If[LessEqual[l, 8e-11], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.9e+98], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6000000000000001e73 or 4.89999999999999979e98 < 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 98.1%

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

   3. Taylor expanded in l around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

   5. Simplified 98.1%

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

   if -2.6000000000000001e73 < l < -3.8e25 or 7.99999999999999952e-11 < l < 4.89999999999999979e98

   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%

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

   if -3.8e25 < l < 7.99999999999999952e-11

   1. Initial program 65.3%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
   (+ U (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.1) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + (J * ((pow(l, 3.0) * 0.3333333333333333) + (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)) <= 0.1d0) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = u + (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (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)) <= 0.1) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.1:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = U + (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.1)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + 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)) <= 0.1)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = U + (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.1], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.10000000000000001

   1. Initial program 82.7%

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

   2. Taylor expanded in l around 0 60.7%

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

      1. associate-*r* 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-*l* 60.8%

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

   4. Simplified 60.8%

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

   if 0.10000000000000001 < (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 88.9%

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

   3. Taylor expanded in K around 0 83.4%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+25} \lor \neg \left(\ell \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.8e+25) (not (<= l 8e-11)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.8e+25) || !(l <= 8e-11)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} 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 <= (-3.8d+25)) .or. (.not. (l <= 8d-11))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    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 <= -3.8e+25) || !(l <= 8e-11)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} 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 <= -3.8e+25) or not (l <= 8e-11):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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 <= -3.8e+25) || !(l <= 8e-11))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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 <= -3.8e+25) || ~((l <= 8e-11)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.8e+25], N[Not[LessEqual[l, 8e-11]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $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 2 regimes

2. if l < -3.8e25 or 7.99999999999999952e-11 < 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 K around 0 74.2%

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

   if -3.8e25 < l < 7.99999999999999952e-11

   1. Initial program 65.3%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+25} \lor \neg \left(\ell \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 6: 78.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+80} \lor \neg \left(\ell \leq 52000000000000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.05e+80) (not (<= l 52000000000000.0)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+80) || !(l <= 52000000000000.0)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} 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 <= (-1.05d+80)) .or. (.not. (l <= 52000000000000.0d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    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 <= -1.05e+80) || !(l <= 52000000000000.0)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} 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 <= -1.05e+80) or not (l <= 52000000000000.0):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	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 <= -1.05e+80) || !(l <= 52000000000000.0))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	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 <= -1.05e+80) || ~((l <= 52000000000000.0)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.05e+80], N[Not[LessEqual[l, 52000000000000.0]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $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 2 regimes

2. if l < -1.05000000000000001e80 or 5.2e13 < 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 88.7%

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

   3. Taylor expanded in l around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*r* 88.7%

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

   5. Simplified 88.7%

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

   6. Taylor expanded in K around 0 67.7%

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

   if -1.05000000000000001e80 < l < 5.2e13

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 86.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+80} \lor \neg \left(\ell \leq 52000000000000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 72.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{+24}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 2.45:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -7.8e+63)
     t_0
     (if (<= l -7.4e+24)
       (* U (- U -4.0))
       (if (<= l 2.45) (+ U (* l (* J 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -7.8e+63) {
		tmp = t_0;
	} else if (l <= -7.4e+24) {
		tmp = U * (U - -4.0);
	} else if (l <= 2.45) {
		tmp = U + (l * (J * 2.0));
	} 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) :: tmp
    t_0 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-7.8d+63)) then
        tmp = t_0
    else if (l <= (-7.4d+24)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 2.45d0) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -7.8e+63) {
		tmp = t_0;
	} else if (l <= -7.4e+24) {
		tmp = U * (U - -4.0);
	} else if (l <= 2.45) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -7.8e+63:
		tmp = t_0
	elif l <= -7.4e+24:
		tmp = U * (U - -4.0)
	elif l <= 2.45:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -7.8e+63)
		tmp = t_0;
	elseif (l <= -7.4e+24)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 2.45)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -7.8e+63)
		tmp = t_0;
	elseif (l <= -7.4e+24)
		tmp = U * (U - -4.0);
	elseif (l <= 2.45)
		tmp = U + (l * (J * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.8e+63], t$95$0, If[LessEqual[l, -7.4e+24], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.8e63 or 2.4500000000000002 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 86.0%

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

   3. Taylor expanded in l around inf 86.0%

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

      1. *-commutative 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*r* 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in K around 0 65.2%

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

   if -7.8e63 < l < -7.39999999999999998e24

   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 37.7%

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

   if -7.39999999999999998e24 < l < 2.4500000000000002

   1. Initial program 65.9%

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

   2. Taylor expanded in l around 0 98.5%

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

   3. Taylor expanded in l around 0 98.5%

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

      1. associate-*r* 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in K around 0 80.2%

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

      1. associate-*r* 80.2%

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

      2. *-commutative 80.2%

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

      3. *-commutative 80.2%

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

   8. Simplified 80.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+63}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{+24}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 2.45:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 42.4% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+19} \lor \neg \left(\ell \leq 1.2\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6e+19) (not (<= l 1.2))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6e+19) || !(l <= 1.2)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6d+19)) .or. (.not. (l <= 1.2d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6e+19) || !(l <= 1.2)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6e+19) or not (l <= 1.2):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6e+19) || !(l <= 1.2))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6e+19) || ~((l <= 1.2)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6e+19], N[Not[LessEqual[l, 1.2]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6e19 or 1.19999999999999996 < 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 16.9%

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

   if -6e19 < l < 1.19999999999999996

   1. Initial program 65.6%

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

   2. Taylor expanded in J around 0 63.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+19} \lor \neg \left(\ell \leq 1.2\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9: 42.4% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+18} \lor \neg \left(\ell \leq 1.05 \cdot 10^{+14}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -8.2e+18) (not (<= l 1.05e+14))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8.2e+18) || !(l <= 1.05e+14)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-8.2d+18)) .or. (.not. (l <= 1.05d+14))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8.2e+18) || !(l <= 1.05e+14)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -8.2e+18) or not (l <= 1.05e+14):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -8.2e+18) || !(l <= 1.05e+14))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -8.2e+18) || ~((l <= 1.05e+14)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -8.2e+18], N[Not[LessEqual[l, 1.05e+14]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -8.2e18 or 1.05e14 < 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 17.1%

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

   if -8.2e18 < l < 1.05e14

   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 61.9%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+18} \lor \neg \left(\ell \leq 1.05 \cdot 10^{+14}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 10: 54.4% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

2. Taylor expanded in l around 0 88.0%

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

3. Taylor expanded in l around 0 61.6%

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

   1. associate-*r* 61.6%

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

5. Simplified 61.6%

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

6. Taylor expanded in K around 0 49.1%

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

   1. associate-*r* 49.1%

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

   2. *-commutative 49.1%

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

   3. *-commutative 49.1%

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

8. Simplified 49.1%

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

9. Final simplification 49.1%

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

Alternative 11: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

5. Simplified 2.6%

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

6. Final simplification 2.6%

   \[\leadsto 1 \]

Alternative 12: 37.4% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

2. Taylor expanded in J around 0 32.3%

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

3. Final simplification 32.3%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023331 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))