Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.4%

Time: 14.6s

Alternatives: 19

Speedup: 1.4×

• 49.7% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* 2.0 (* J (log1p (expm1 (* l (cos (* K 0.5))))))) U))

C

double code(double J, double l, double K, double U) {
	return (2.0 * (J * log1p(expm1((l * cos((K * 0.5))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (2.0 * (J * Math.log1p(Math.expm1((l * Math.cos((K * 0.5))))))) + U;
}

Python

def code(J, l, K, U):
	return (2.0 * (J * math.log1p(math.expm1((l * math.cos((K * 0.5))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(2.0 * Float64(J * log1p(expm1(Float64(l * cos(Float64(K * 0.5))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Log[1 + N[(Exp[N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in l around 0 71.1%

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

   1. log1p-expm1-u 99.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   2. *-commutative 99.6%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

4. Applied egg-rr 99.6%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

5. Final simplification 99.6%

   \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \]

Alternative 2: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;U + \left(\left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq -110:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.00069 \lor \neg \left(\ell \leq 2.9 \cdot 10^{+84}\right):\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* J (- (exp l) (exp (- l))))))
   (if (<= l -1.9e+123)
     (+ U (* (* (* J 0.3333333333333333) (pow l 3.0)) t_0))
     (if (<= l -110.0)
       t_1
       (if (or (<= l 0.00069) (not (<= l 2.9e+84)))
         (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* 2.0 l)))))
         (+ U t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * (exp(l) - exp(-l));
	double tmp;
	if (l <= -1.9e+123) {
		tmp = U + (((J * 0.3333333333333333) * pow(l, 3.0)) * t_0);
	} else if (l <= -110.0) {
		tmp = t_1;
	} else if ((l <= 0.00069) || !(l <= 2.9e+84)) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (2.0 * l))));
	} else {
		tmp = U + 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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = j * (exp(l) - exp(-l))
    if (l <= (-1.9d+123)) then
        tmp = u + (((j * 0.3333333333333333d0) * (l ** 3.0d0)) * t_0)
    else if (l <= (-110.0d0)) then
        tmp = t_1
    else if ((l <= 0.00069d0) .or. (.not. (l <= 2.9d+84))) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (2.0d0 * l))))
    else
        tmp = u + 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 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if (l <= -1.9e+123) {
		tmp = U + (((J * 0.3333333333333333) * Math.pow(l, 3.0)) * t_0);
	} else if (l <= -110.0) {
		tmp = t_1;
	} else if ((l <= 0.00069) || !(l <= 2.9e+84)) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (2.0 * l))));
	} else {
		tmp = U + t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if l <= -1.9e+123:
		tmp = U + (((J * 0.3333333333333333) * math.pow(l, 3.0)) * t_0)
	elif l <= -110.0:
		tmp = t_1
	elif (l <= 0.00069) or not (l <= 2.9e+84):
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (2.0 * l))))
	else:
		tmp = U + t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (l <= -1.9e+123)
		tmp = Float64(U + Float64(Float64(Float64(J * 0.3333333333333333) * (l ^ 3.0)) * t_0));
	elseif (l <= -110.0)
		tmp = t_1;
	elseif ((l <= 0.00069) || !(l <= 2.9e+84))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(2.0 * l)))));
	else
		tmp = Float64(U + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if (l <= -1.9e+123)
		tmp = U + (((J * 0.3333333333333333) * (l ^ 3.0)) * t_0);
	elseif (l <= -110.0)
		tmp = t_1;
	elseif ((l <= 0.00069) || ~((l <= 2.9e+84)))
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (2.0 * l))));
	else
		tmp = U + 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[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+123], N[(U + N[(N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -110.0], t$95$1, If[Or[LessEqual[l, 0.00069], N[Not[LessEqual[l, 2.9e+84]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.89999999999999997e123

   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%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if -1.89999999999999997e123 < l < -110

   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 81.5%

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

   3. Taylor expanded in J around inf 81.5%

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

   if -110 < l < 6.89999999999999967e-4 or 2.89999999999999989e84 < l

   1. Initial program 79.6%

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

   2. Taylor expanded in l around 0 98.8%

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

   if 6.89999999999999967e-4 < l < 2.89999999999999989e84

   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 73.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;U + \left(\left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -110:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.00069 \lor \neg \left(\ell \leq 2.9 \cdot 10^{+84}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left(\left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\right) \cdot t_0\\ t_2 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00035:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+84}:\\ \;\;\;\;U + t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* (* (* J 0.3333333333333333) (pow l 3.0)) t_0)))
        (t_2 (* J (- (exp l) (exp (- l))))))
   (if (<= l -1.9e+123)
     t_1
     (if (<= l -0.00035)
       t_2
       (if (<= l 4.4e-47)
         (+ U (* t_0 (* l (* 2.0 J))))
         (if (<= l 2.9e+84) (+ U 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 + (((J * 0.3333333333333333) * pow(l, 3.0)) * t_0);
	double t_2 = J * (exp(l) - exp(-l));
	double tmp;
	if (l <= -1.9e+123) {
		tmp = t_1;
	} else if (l <= -0.00035) {
		tmp = t_2;
	} else if (l <= 4.4e-47) {
		tmp = U + (t_0 * (l * (2.0 * J)));
	} else if (l <= 2.9e+84) {
		tmp = U + 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 + (((j * 0.3333333333333333d0) * (l ** 3.0d0)) * t_0)
    t_2 = j * (exp(l) - exp(-l))
    if (l <= (-1.9d+123)) then
        tmp = t_1
    else if (l <= (-0.00035d0)) then
        tmp = t_2
    else if (l <= 4.4d-47) then
        tmp = u + (t_0 * (l * (2.0d0 * j)))
    else if (l <= 2.9d+84) then
        tmp = u + 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 + (((J * 0.3333333333333333) * Math.pow(l, 3.0)) * t_0);
	double t_2 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if (l <= -1.9e+123) {
		tmp = t_1;
	} else if (l <= -0.00035) {
		tmp = t_2;
	} else if (l <= 4.4e-47) {
		tmp = U + (t_0 * (l * (2.0 * J)));
	} else if (l <= 2.9e+84) {
		tmp = U + 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 + (((J * 0.3333333333333333) * math.pow(l, 3.0)) * t_0)
	t_2 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if l <= -1.9e+123:
		tmp = t_1
	elif l <= -0.00035:
		tmp = t_2
	elif l <= 4.4e-47:
		tmp = U + (t_0 * (l * (2.0 * J)))
	elif l <= 2.9e+84:
		tmp = U + 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(Float64(Float64(J * 0.3333333333333333) * (l ^ 3.0)) * t_0))
	t_2 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (l <= -1.9e+123)
		tmp = t_1;
	elseif (l <= -0.00035)
		tmp = t_2;
	elseif (l <= 4.4e-47)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(2.0 * J))));
	elseif (l <= 2.9e+84)
		tmp = Float64(U + 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 + (((J * 0.3333333333333333) * (l ^ 3.0)) * t_0);
	t_2 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if (l <= -1.9e+123)
		tmp = t_1;
	elseif (l <= -0.00035)
		tmp = t_2;
	elseif (l <= 4.4e-47)
		tmp = U + (t_0 * (l * (2.0 * J)));
	elseif (l <= 2.9e+84)
		tmp = U + 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[(N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+123], t$95$1, If[LessEqual[l, -0.00035], t$95$2, If[LessEqual[l, 4.4e-47], N[(U + N[(t$95$0 * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e+84], N[(U + t$95$2), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.89999999999999997e123 or 2.89999999999999989e84 < 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 97.6%

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

   3. Taylor expanded in l around inf 97.6%

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

      1. associate-*r* 97.6%

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

   5. Simplified 97.6%

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

   if -1.89999999999999997e123 < l < -3.49999999999999996e-4

   1. Initial program 99.4%

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

   2. Taylor expanded in K around 0 81.6%

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

   3. Taylor expanded in J around inf 81.6%

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

   if -3.49999999999999996e-4 < l < 4.40000000000000037e-47

   1. Initial program 71.1%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 4.40000000000000037e-47 < l < 2.89999999999999989e84

   1. Initial program 99.3%

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

   2. Taylor expanded in K around 0 83.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;U + \left(\left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -0.00035:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+84}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 4: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.522:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.522)
     (+ U (* t_0 (* l (* 2.0 J))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* 2.0 l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.522) {
		tmp = U + (t_0 * (l * (2.0 * J)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (2.0 * l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.522d0) then
        tmp = u + (t_0 * (l * (2.0d0 * j)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (2.0d0 * l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.522) {
		tmp = U + (t_0 * (l * (2.0 * J)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (2.0 * l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.522:
		tmp = U + (t_0 * (l * (2.0 * J)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (2.0 * l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.522)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(2.0 * J))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(2.0 * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.522)
		tmp = U + (t_0 * (l * (2.0 * J)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (2.0 * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.522], N[(U + N[(t$95$0 * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.52200000000000002

   1. Initial program 81.4%

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

   2. Taylor expanded in l around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

   4. Simplified 71.0%

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

   if 0.52200000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 88.2%

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

   2. Taylor expanded in l around 0 89.2%

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

   3. Taylor expanded in K around 0 87.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.522:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\\ \end{array} \]

Alternative 5: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -0.00035:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 220:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= l -0.00035)
     t_0
     (if (<= l 220.0)
       (+ U (* (cos (/ K 2.0)) (* l (* 2.0 J))))
       (if (<= l 2.4e+89)
         t_0
         (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (l <= -0.00035) {
		tmp = t_0;
	} else if (l <= 220.0) {
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 2.4e+89) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    if (l <= (-0.00035d0)) then
        tmp = t_0
    else if (l <= 220.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (2.0d0 * j)))
    else if (l <= 2.4d+89) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if (l <= -0.00035) {
		tmp = t_0;
	} else if (l <= 220.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 2.4e+89) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if l <= -0.00035:
		tmp = t_0
	elif l <= 220.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (2.0 * J)))
	elif l <= 2.4e+89:
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (l <= -0.00035)
		tmp = t_0;
	elseif (l <= 220.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(2.0 * J))));
	elseif (l <= 2.4e+89)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if (l <= -0.00035)
		tmp = t_0;
	elseif (l <= 220.0)
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	elseif (l <= 2.4e+89)
		tmp = t_0;
	else
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -0.00035], t$95$0, If[LessEqual[l, 220.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+89], t$95$0, N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.49999999999999996e-4 or 220 < l < 2.40000000000000004e89

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0 77.3%

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

   3. Taylor expanded in J around inf 77.3%

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

   if -3.49999999999999996e-4 < l < 220

   1. Initial program 73.4%

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

   2. Taylor expanded in l around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   if 2.40000000000000004e89 < 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 48.8%

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

   3. Taylor expanded in K around 0 67.1%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00035:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 220:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+89}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 6: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{+80} \lor \neg \left(J \leq 5.8 \cdot 10^{+43}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -2.6e+80) (not (<= J 5.8e+43)))
   (+ U (* (cos (/ K 2.0)) (* l (* 2.0 J))))
   (+ U (* J (- (exp l) (exp (- l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -2.6e+80) || !(J <= 5.8e+43)) {
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	} else {
		tmp = U + (J * (exp(l) - exp(-l)));
	}
	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 ((j <= (-2.6d+80)) .or. (.not. (j <= 5.8d+43))) then
        tmp = u + (cos((k / 2.0d0)) * (l * (2.0d0 * j)))
    else
        tmp = u + (j * (exp(l) - exp(-l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -2.6e+80) || !(J <= 5.8e+43)) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (2.0 * J)));
	} else {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -2.6e+80) or not (J <= 5.8e+43):
		tmp = U + (math.cos((K / 2.0)) * (l * (2.0 * J)))
	else:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -2.6e+80) || !(J <= 5.8e+43))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(2.0 * J))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -2.6e+80) || ~((J <= 5.8e+43)))
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	else
		tmp = U + (J * (exp(l) - exp(-l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -2.6e+80], N[Not[LessEqual[J, 5.8e+43]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -2.59999999999999982e80 or 5.8000000000000004e43 < J

   1. Initial program 73.1%

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

   2. Taylor expanded in l around 0 92.7%

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

      1. associate-*r* 92.7%

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

      2. *-commutative 92.7%

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

   4. Simplified 92.7%

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

   if -2.59999999999999982e80 < J < 5.8000000000000004e43

   1. Initial program 94.3%

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

   2. Taylor expanded in K around 0 82.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{+80} \lor \neg \left(J \leq 5.8 \cdot 10^{+43}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \]

Alternative 7: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -4150000000000:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107} \lor \neg \left(\ell \leq 1.5 \cdot 10^{+261}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0)))))
   (if (<= l -4e+91)
     t_0
     (if (<= l -4150000000000.0)
       (fabs (* U (+ U 4.0)))
       (if (or (<= l 1.2e+107) (not (<= l 1.5e+261)))
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -4150000000000.0) {
		tmp = fabs((U * (U + 4.0)));
	} else if ((l <= 1.2e+107) || !(l <= 1.5e+261)) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} 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 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    if (l <= (-4d+91)) then
        tmp = t_0
    else if (l <= (-4150000000000.0d0)) then
        tmp = abs((u * (u + 4.0d0)))
    else if ((l <= 1.2d+107) .or. (.not. (l <= 1.5d+261))) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    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 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -4150000000000.0) {
		tmp = Math.abs((U * (U + 4.0)));
	} else if ((l <= 1.2e+107) || !(l <= 1.5e+261)) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	tmp = 0
	if l <= -4e+91:
		tmp = t_0
	elif l <= -4150000000000.0:
		tmp = math.fabs((U * (U + 4.0)))
	elif (l <= 1.2e+107) or not (l <= 1.5e+261):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -4150000000000.0)
		tmp = abs(Float64(U * Float64(U + 4.0)));
	elseif ((l <= 1.2e+107) || !(l <= 1.5e+261))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (J * (l ^ 3.0));
	tmp = 0.0;
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -4150000000000.0)
		tmp = abs((U * (U + 4.0)));
	elseif ((l <= 1.2e+107) || ~((l <= 1.5e+261)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e+91], t$95$0, If[LessEqual[l, -4150000000000.0], N[Abs[N[(U * N[(U + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 1.2e+107], N[Not[LessEqual[l, 1.5e+261]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.00000000000000032e91 or 1.2e107 < l < 1.49999999999999989e261

   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 98.7%

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

   3. Taylor expanded in K around 0 74.0%

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

   4. Taylor expanded in l around inf 74.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   if -4.00000000000000032e91 < l < -4.15e12

   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 39.3%

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

      1. add-sqr-sqrt 39.2%

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

      2. sqrt-unprod 39.3%

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

      3. pow2 39.3%

         \[\leadsto \sqrt{\color{blue}{{\left(U \cdot \left(U - -4\right)\right)}^{2}}} \]

      4. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(1 \cdot \left(U - -4\right)\right)}\right)}^{2}} \]

      5. sub-neg 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(1 \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)\right)}^{2}} \]

      6. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)}^{2}} \]

      7. metadata-eval 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(U + \color{blue}{4}\right)\right)}^{2}} \]

   5. Applied egg-rr 39.3%

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

      1. unpow2 39.3%

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

      2. rem-sqrt-square 39.4%

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

      3. +-commutative 39.4%

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

   7. Simplified 39.4%

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

   if -4.15e12 < l < 1.2e107 or 1.49999999999999989e261 < l

   1. Initial program 78.8%

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

   2. Taylor expanded in l around 0 85.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -4150000000000:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107} \lor \neg \left(\ell \leq 1.5 \cdot 10^{+261}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -14600000000000:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 \cdot J\right) \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+261}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0)))) (t_1 (cos (* K 0.5))))
   (if (<= l -4e+91)
     t_0
     (if (<= l -14600000000000.0)
       (fabs (* U (+ U 4.0)))
       (if (<= l 1.2e+107)
         (+ U (* l (* (* 2.0 J) t_1)))
         (if (<= l 1.5e+261) t_0 (+ U (* 2.0 (* J (* l t_1))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -14600000000000.0) {
		tmp = fabs((U * (U + 4.0)));
	} else if (l <= 1.2e+107) {
		tmp = U + (l * ((2.0 * J) * t_1));
	} else if (l <= 1.5e+261) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * 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) :: tmp
    t_0 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    t_1 = cos((k * 0.5d0))
    if (l <= (-4d+91)) then
        tmp = t_0
    else if (l <= (-14600000000000.0d0)) then
        tmp = abs((u * (u + 4.0d0)))
    else if (l <= 1.2d+107) then
        tmp = u + (l * ((2.0d0 * j) * t_1))
    else if (l <= 1.5d+261) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (j * (l * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -14600000000000.0) {
		tmp = Math.abs((U * (U + 4.0)));
	} else if (l <= 1.2e+107) {
		tmp = U + (l * ((2.0 * J) * t_1));
	} else if (l <= 1.5e+261) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * t_1)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -4e+91:
		tmp = t_0
	elif l <= -14600000000000.0:
		tmp = math.fabs((U * (U + 4.0)))
	elif l <= 1.2e+107:
		tmp = U + (l * ((2.0 * J) * t_1))
	elif l <= 1.5e+261:
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l * t_1)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -14600000000000.0)
		tmp = abs(Float64(U * Float64(U + 4.0)));
	elseif (l <= 1.2e+107)
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 * J) * t_1)));
	elseif (l <= 1.5e+261)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (J * (l ^ 3.0));
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -14600000000000.0)
		tmp = abs((U * (U + 4.0)));
	elseif (l <= 1.2e+107)
		tmp = U + (l * ((2.0 * J) * t_1));
	elseif (l <= 1.5e+261)
		tmp = t_0;
	else
		tmp = U + (2.0 * (J * (l * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e+91], t$95$0, If[LessEqual[l, -14600000000000.0], N[Abs[N[(U * N[(U + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.2e+107], N[(U + N[(l * N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+261], t$95$0, N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.00000000000000032e91 or 1.2e107 < l < 1.49999999999999989e261

   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 98.7%

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

   3. Taylor expanded in K around 0 74.0%

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

   4. Taylor expanded in l around inf 74.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   if -4.00000000000000032e91 < l < -1.46e13

   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 39.3%

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

      1. add-sqr-sqrt 39.2%

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

      2. sqrt-unprod 39.3%

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

      3. pow2 39.3%

         \[\leadsto \sqrt{\color{blue}{{\left(U \cdot \left(U - -4\right)\right)}^{2}}} \]

      4. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(1 \cdot \left(U - -4\right)\right)}\right)}^{2}} \]

      5. sub-neg 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(1 \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)\right)}^{2}} \]

      6. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)}^{2}} \]

      7. metadata-eval 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(U + \color{blue}{4}\right)\right)}^{2}} \]

   5. Applied egg-rr 39.3%

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

      1. unpow2 39.3%

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

      2. rem-sqrt-square 39.4%

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

      3. +-commutative 39.4%

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

   7. Simplified 39.4%

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

   if -1.46e13 < l < 1.2e107

   1. Initial program 77.3%

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

   2. Taylor expanded in l around 0 86.7%

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

      1. associate-*r* 86.7%

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

      2. *-commutative 86.7%

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

      3. associate-*l* 86.7%

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

   4. Simplified 86.7%

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

   if 1.49999999999999989e261 < 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 73.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -14600000000000:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+261}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 9: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{+14}:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+261}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0)))))
   (if (<= l -4e+91)
     t_0
     (if (<= l -8.4e+14)
       (fabs (* U (+ U 4.0)))
       (if (<= l 1.2e+107)
         (+ U (* (cos (/ K 2.0)) (* l (* 2.0 J))))
         (if (<= l 1.35e+261)
           t_0
           (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -8.4e+14) {
		tmp = fabs((U * (U + 4.0)));
	} else if (l <= 1.2e+107) {
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 1.35e+261) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    if (l <= (-4d+91)) then
        tmp = t_0
    else if (l <= (-8.4d+14)) then
        tmp = abs((u * (u + 4.0d0)))
    else if (l <= 1.2d+107) then
        tmp = u + (cos((k / 2.0d0)) * (l * (2.0d0 * j)))
    else if (l <= 1.35d+261) then
        tmp = t_0
    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 t_0 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -8.4e+14) {
		tmp = Math.abs((U * (U + 4.0)));
	} else if (l <= 1.2e+107) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 1.35e+261) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	tmp = 0
	if l <= -4e+91:
		tmp = t_0
	elif l <= -8.4e+14:
		tmp = math.fabs((U * (U + 4.0)))
	elif l <= 1.2e+107:
		tmp = U + (math.cos((K / 2.0)) * (l * (2.0 * J)))
	elif l <= 1.35e+261:
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -8.4e+14)
		tmp = abs(Float64(U * Float64(U + 4.0)));
	elseif (l <= 1.2e+107)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(2.0 * J))));
	elseif (l <= 1.35e+261)
		tmp = t_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)
	t_0 = 0.3333333333333333 * (J * (l ^ 3.0));
	tmp = 0.0;
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -8.4e+14)
		tmp = abs((U * (U + 4.0)));
	elseif (l <= 1.2e+107)
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	elseif (l <= 1.35e+261)
		tmp = t_0;
	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[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e+91], t$95$0, If[LessEqual[l, -8.4e+14], N[Abs[N[(U * N[(U + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.2e+107], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+261], t$95$0, 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 4 regimes

2. if l < -4.00000000000000032e91 or 1.2e107 < l < 1.35000000000000001e261

   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 98.7%

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

   3. Taylor expanded in K around 0 74.0%

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

   4. Taylor expanded in l around inf 74.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   if -4.00000000000000032e91 < l < -8.4e14

   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 39.3%

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

      1. add-sqr-sqrt 39.2%

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

      2. sqrt-unprod 39.3%

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

      3. pow2 39.3%

         \[\leadsto \sqrt{\color{blue}{{\left(U \cdot \left(U - -4\right)\right)}^{2}}} \]

      4. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(1 \cdot \left(U - -4\right)\right)}\right)}^{2}} \]

      5. sub-neg 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(1 \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)\right)}^{2}} \]

      6. *-un-lft-identity 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)}^{2}} \]

      7. metadata-eval 39.3%

         \[\leadsto \sqrt{{\left(U \cdot \left(U + \color{blue}{4}\right)\right)}^{2}} \]

   5. Applied egg-rr 39.3%

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

      1. unpow2 39.3%

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

      2. rem-sqrt-square 39.4%

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

      3. +-commutative 39.4%

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

   7. Simplified 39.4%

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

   if -8.4e14 < l < 1.2e107

   1. Initial program 77.3%

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

   2. Taylor expanded in l around 0 86.8%

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

      1. associate-*r* 86.8%

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

      2. *-commutative 86.8%

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

   4. Simplified 86.8%

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

   if 1.35000000000000001e261 < 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 73.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{+14}:\\ \;\;\;\;\left|U \cdot \left(U + 4\right)\right|\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+261}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 10: 71.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -170000:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 7600000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0)))))
   (if (<= l -4e+91)
     t_0
     (if (<= l -170000.0)
       (* U (- U -4.0))
       (if (<= l 7600000.0) (fma (* 2.0 J) l U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double tmp;
	if (l <= -4e+91) {
		tmp = t_0;
	} else if (l <= -170000.0) {
		tmp = U * (U - -4.0);
	} else if (l <= 7600000.0) {
		tmp = fma((2.0 * J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -4e+91)
		tmp = t_0;
	elseif (l <= -170000.0)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 7600000.0)
		tmp = fma(Float64(2.0 * J), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e+91], t$95$0, If[LessEqual[l, -170000.0], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7600000.0], N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.00000000000000032e91 or 7.6e6 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 86.1%

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

   3. Taylor expanded in K around 0 59.1%

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

   4. Taylor expanded in l around inf 59.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 59.1%

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

   6. Simplified 59.1%

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

   if -4.00000000000000032e91 < l < -1.7e5

   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 34.5%

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

   if -1.7e5 < l < 7.6e6

   1. Initial program 73.7%

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

   2. Taylor expanded in l around 0 98.6%

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

   3. Taylor expanded in K around 0 84.9%

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

   4. Taylor expanded in J around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. associate-*r* 84.9%

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

      3. fma-def 84.9%

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

   6. Simplified 84.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+91}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -170000:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 7600000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 11: 53.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in l around 0 71.1%

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

3. Taylor expanded in K around 0 57.7%

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

4. Taylor expanded in J around 0 57.7%

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

   1. +-commutative 57.7%

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

   2. associate-*r* 57.7%

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

   3. fma-def 57.7%

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

6. Simplified 57.7%

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

7. Final simplification 57.7%

   \[\leadsto \mathsf{fma}\left(2 \cdot J, \ell, U\right) \]

Alternative 12: 42.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \left(16 - U\right)\\ \mathbf{elif}\;\ell \leq -132000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U + -16\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.4e+110)
   (* U (- 16.0 U))
   (if (<= l -132000.0) (* U U) (if (<= l 1.4e+70) U (* U (+ U -16.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.4e+110) {
		tmp = U * (16.0 - U);
	} else if (l <= -132000.0) {
		tmp = U * U;
	} else if (l <= 1.4e+70) {
		tmp = U;
	} else {
		tmp = U * (U + -16.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-3.4d+110)) then
        tmp = u * (16.0d0 - u)
    else if (l <= (-132000.0d0)) then
        tmp = u * u
    else if (l <= 1.4d+70) then
        tmp = u
    else
        tmp = u * (u + (-16.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.4e+110) {
		tmp = U * (16.0 - U);
	} else if (l <= -132000.0) {
		tmp = U * U;
	} else if (l <= 1.4e+70) {
		tmp = U;
	} else {
		tmp = U * (U + -16.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.4e+110:
		tmp = U * (16.0 - U)
	elif l <= -132000.0:
		tmp = U * U
	elif l <= 1.4e+70:
		tmp = U
	else:
		tmp = U * (U + -16.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.4e+110)
		tmp = Float64(U * Float64(16.0 - U));
	elseif (l <= -132000.0)
		tmp = Float64(U * U);
	elseif (l <= 1.4e+70)
		tmp = U;
	else
		tmp = Float64(U * Float64(U + -16.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.4e+110)
		tmp = U * (16.0 - U);
	elseif (l <= -132000.0)
		tmp = U * U;
	elseif (l <= 1.4e+70)
		tmp = U;
	else
		tmp = U * (U + -16.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.4e+110], N[(U * N[(16.0 - U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -132000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.4e+70], U, N[(U * N[(U + -16.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.4000000000000001e110

   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 53.3%

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

   3. Taylor expanded in K around 0 41.4%

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

   5. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{U \cdot \left(-\left(U + -16\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 22.3%

         \[\leadsto U \cdot \left(-\color{blue}{\left(-16 + U\right)}\right) \]

      2. distribute-neg-in 22.3%

         \[\leadsto U \cdot \color{blue}{\left(\left(--16\right) + \left(-U\right)\right)} \]

      3. metadata-eval 22.3%

         \[\leadsto U \cdot \left(\color{blue}{16} + \left(-U\right)\right) \]

      4. sub-neg 22.3%

         \[\leadsto U \cdot \color{blue}{\left(16 - U\right)} \]

   7. Simplified 22.3%

      \[\leadsto \color{blue}{U \cdot \left(16 - U\right)} \]

   if -3.4000000000000001e110 < l < -132000

   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 31.6%

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

   if -132000 < l < 1.39999999999999995e70

   1. Initial program 75.8%

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

   2. Taylor expanded in J around 0 65.8%

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

   if 1.39999999999999995e70 < 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 48.0%

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

   3. Taylor expanded in K around 0 24.3%

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

   5. Applied egg-rr 15.5%

      \[\leadsto \color{blue}{\left(U + -16\right) \cdot U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \left(16 - U\right)\\ \mathbf{elif}\;\ell \leq -132000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U + -16\right)\\ \end{array} \]

Alternative 13: 42.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \frac{U}{-3}\\ \mathbf{elif}\;\ell \leq -132000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+66}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U + -16\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.7e+110)
   (* U (/ U -3.0))
   (if (<= l -132000.0) (* U U) (if (<= l 5.7e+66) U (* U (+ U -16.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.7e+110) {
		tmp = U * (U / -3.0);
	} else if (l <= -132000.0) {
		tmp = U * U;
	} else if (l <= 5.7e+66) {
		tmp = U;
	} else {
		tmp = U * (U + -16.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-3.7d+110)) then
        tmp = u * (u / (-3.0d0))
    else if (l <= (-132000.0d0)) then
        tmp = u * u
    else if (l <= 5.7d+66) then
        tmp = u
    else
        tmp = u * (u + (-16.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.7e+110) {
		tmp = U * (U / -3.0);
	} else if (l <= -132000.0) {
		tmp = U * U;
	} else if (l <= 5.7e+66) {
		tmp = U;
	} else {
		tmp = U * (U + -16.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.7e+110:
		tmp = U * (U / -3.0)
	elif l <= -132000.0:
		tmp = U * U
	elif l <= 5.7e+66:
		tmp = U
	else:
		tmp = U * (U + -16.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.7e+110)
		tmp = Float64(U * Float64(U / -3.0));
	elseif (l <= -132000.0)
		tmp = Float64(U * U);
	elseif (l <= 5.7e+66)
		tmp = U;
	else
		tmp = Float64(U * Float64(U + -16.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.7e+110)
		tmp = U * (U / -3.0);
	elseif (l <= -132000.0)
		tmp = U * U;
	elseif (l <= 5.7e+66)
		tmp = U;
	else
		tmp = U * (U + -16.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.7e+110], N[(U * N[(U / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -132000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 5.7e+66], U, N[(U * N[(U + -16.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.70000000000000012e110

   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 53.3%

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

   3. Taylor expanded in K around 0 41.4%

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

   5. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\frac{U}{-3} \cdot U} \]

   if -3.70000000000000012e110 < l < -132000

   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 31.6%

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

   if -132000 < l < 5.7000000000000003e66

   1. Initial program 75.8%

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

   2. Taylor expanded in J around 0 65.8%

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

   if 5.7000000000000003e66 < 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 48.0%

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

   3. Taylor expanded in K around 0 24.3%

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

   5. Applied egg-rr 15.5%

      \[\leadsto \color{blue}{\left(U + -16\right) \cdot U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \frac{U}{-3}\\ \mathbf{elif}\;\ell \leq -132000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+66}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U + -16\right)\\ \end{array} \]

Alternative 14: 42.2% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \left(16 - U\right)\\ \mathbf{elif}\;\ell \leq -132000 \lor \neg \left(\ell \leq 3.1 \cdot 10^{+66}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.3e+110)
   (* U (- 16.0 U))
   (if (or (<= l -132000.0) (not (<= l 3.1e+66))) (* U U) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.3e+110) {
		tmp = U * (16.0 - U);
	} else if ((l <= -132000.0) || !(l <= 3.1e+66)) {
		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 <= (-4.3d+110)) then
        tmp = u * (16.0d0 - u)
    else if ((l <= (-132000.0d0)) .or. (.not. (l <= 3.1d+66))) 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 <= -4.3e+110) {
		tmp = U * (16.0 - U);
	} else if ((l <= -132000.0) || !(l <= 3.1e+66)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.3e+110:
		tmp = U * (16.0 - U)
	elif (l <= -132000.0) or not (l <= 3.1e+66):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.3e+110)
		tmp = Float64(U * Float64(16.0 - U));
	elseif ((l <= -132000.0) || !(l <= 3.1e+66))
		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 <= -4.3e+110)
		tmp = U * (16.0 - U);
	elseif ((l <= -132000.0) || ~((l <= 3.1e+66)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.3e+110], N[(U * N[(16.0 - U), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -132000.0], N[Not[LessEqual[l, 3.1e+66]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]]

Derivation

1. Split input into 3 regimes

2. if l < -4.30000000000000007e110

   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 53.3%

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

   3. Taylor expanded in K around 0 41.4%

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

   5. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{U \cdot \left(-\left(U + -16\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 22.3%

         \[\leadsto U \cdot \left(-\color{blue}{\left(-16 + U\right)}\right) \]

      2. distribute-neg-in 22.3%

         \[\leadsto U \cdot \color{blue}{\left(\left(--16\right) + \left(-U\right)\right)} \]

      3. metadata-eval 22.3%

         \[\leadsto U \cdot \left(\color{blue}{16} + \left(-U\right)\right) \]

      4. sub-neg 22.3%

         \[\leadsto U \cdot \color{blue}{\left(16 - U\right)} \]

   7. Simplified 22.3%

      \[\leadsto \color{blue}{U \cdot \left(16 - U\right)} \]

   if -4.30000000000000007e110 < l < -132000 or 3.10000000000000019e66 < 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 20.5%

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

   if -132000 < l < 3.10000000000000019e66

   1. Initial program 75.8%

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

   2. Taylor expanded in J around 0 65.8%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+110}:\\ \;\;\;\;U \cdot \left(16 - U\right)\\ \mathbf{elif}\;\ell \leq -132000 \lor \neg \left(\ell \leq 3.1 \cdot 10^{+66}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 15: 42.0% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -155000 \lor \neg \left(\ell \leq 3.1 \cdot 10^{+66}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -155000.0) (not (<= l 3.1e+66))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -155000.0) || !(l <= 3.1e+66)) {
		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 <= (-155000.0d0)) .or. (.not. (l <= 3.1d+66))) 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 <= -155000.0) || !(l <= 3.1e+66)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -155000.0) or not (l <= 3.1e+66):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -155000.0) || !(l <= 3.1e+66))
		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 <= -155000.0) || ~((l <= 3.1e+66)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -155000.0], N[Not[LessEqual[l, 3.1e+66]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -155000 or 3.10000000000000019e66 < 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 14.7%

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

   if -155000 < l < 3.10000000000000019e66

   1. Initial program 75.8%

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

   2. Taylor expanded in J around 0 65.8%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -155000 \lor \neg \left(\ell \leq 3.1 \cdot 10^{+66}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 16: 53.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 * (j * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in l around 0 71.1%

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

3. Taylor expanded in K around 0 57.7%

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

4. Final simplification 57.7%

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

Alternative 17: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -15 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -15.0)

C

double code(double J, double l, double K, double U) {
	return -15.0;
}

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 = -15.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -15.0;
}

Python

def code(J, l, K, U):
	return -15.0

Julia

function code(J, l, K, U)
	return -15.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -15.0;
end

Wolfram

code[J_, l_, K_, U_] := -15.0

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in l around 0 71.1%

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

3. Taylor expanded in K around 0 57.7%

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

5. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{\frac{-3}{U + -64} \cdot \left(\left(U + 16\right) - U \cdot -4\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 2.9%

      \[\leadsto \color{blue}{\frac{-3 \cdot \left(\left(U + 16\right) - U \cdot -4\right)}{U + -64}} \]

   2. +-commutative 2.9%

      \[\leadsto \frac{-3 \cdot \left(\color{blue}{\left(16 + U\right)} - U \cdot -4\right)}{U + -64} \]

   3. associate-+r- 2.9%

      \[\leadsto \frac{-3 \cdot \color{blue}{\left(16 + \left(U - U \cdot -4\right)\right)}}{U + -64} \]

   4. distribute-lft-in 2.9%

      \[\leadsto \frac{\color{blue}{-3 \cdot 16 + -3 \cdot \left(U - U \cdot -4\right)}}{U + -64} \]

   5. metadata-eval 2.9%

      \[\leadsto \frac{\color{blue}{-48} + -3 \cdot \left(U - U \cdot -4\right)}{U + -64} \]

   6. *-commutative 2.9%

      \[\leadsto \frac{-48 + -3 \cdot \left(U - \color{blue}{-4 \cdot U}\right)}{U + -64} \]

   7. cancel-sign-sub-inv 2.9%

      \[\leadsto \frac{-48 + -3 \cdot \color{blue}{\left(U + \left(--4\right) \cdot U\right)}}{U + -64} \]

   8. metadata-eval 2.9%

      \[\leadsto \frac{-48 + -3 \cdot \left(U + \color{blue}{4} \cdot U\right)}{U + -64} \]

   9. distribute-lft-in 2.9%

      \[\leadsto \frac{-48 + \color{blue}{\left(-3 \cdot U + -3 \cdot \left(4 \cdot U\right)\right)}}{U + -64} \]

   10. associate-*r* 2.9%

       \[\leadsto \frac{-48 + \left(-3 \cdot U + \color{blue}{\left(-3 \cdot 4\right) \cdot U}\right)}{U + -64} \]

   11. distribute-rgt-out 2.9%

       \[\leadsto \frac{-48 + \color{blue}{U \cdot \left(-3 + -3 \cdot 4\right)}}{U + -64} \]

   12. metadata-eval 2.9%

       \[\leadsto \frac{-48 + U \cdot \left(-3 + \color{blue}{-12}\right)}{U + -64} \]

   13. metadata-eval 2.9%

       \[\leadsto \frac{-48 + U \cdot \color{blue}{-15}}{U + -64} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{\frac{-48 + U \cdot -15}{U + -64}} \]

8. Taylor expanded in U around inf 2.9%

   \[\leadsto \color{blue}{-15} \]

9. Final simplification 2.9%

   \[\leadsto -15 \]

Alternative 18: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.3333333333333333)

C

double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

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 = -0.3333333333333333d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

Python

def code(J, l, K, U):
	return -0.3333333333333333

Julia

function code(J, l, K, U)
	return -0.3333333333333333
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.3333333333333333;
end

Wolfram

code[J_, l_, K_, U_] := -0.3333333333333333

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in l around 0 71.1%

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

3. Taylor expanded in K around 0 57.7%

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

5. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{\frac{\frac{U}{-3}}{U}} \]
6. Step-by-step derivation

   1. associate-/l/ 2.9%

      \[\leadsto \color{blue}{\frac{U}{U \cdot -3}} \]

   2. associate-/r* 2.9%

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

   3. *-inverses 2.9%

      \[\leadsto \frac{\color{blue}{1}}{-3} \]

   4. metadata-eval 2.9%

      \[\leadsto \color{blue}{-0.3333333333333333} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{-0.3333333333333333} \]

8. Final simplification 2.9%

   \[\leadsto -0.3333333333333333 \]

Alternative 19: 36.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in J around 0 39.3%

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

3. Final simplification 39.3%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023330 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))