Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.4%

Time: 12.6s

Alternatives: 17

Speedup: 1.4×

• 50.1% 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: 86.5% 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 84.9%

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

3. Taylor expanded in l around 0 58.7%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. 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%

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

5. Applied egg-rr 99.6%

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

Alternative 2: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_0\\ t_2 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5.4e+124)
     t_1
     (if (<= l -14500.0)
       t_2
       (if (<= l 0.39)
         (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 5e+93) 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 <= -5.4e+124) {
		tmp = t_1;
	} else if (l <= -14500.0) {
		tmp = t_2;
	} else if (l <= 0.39) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 5e+93) {
		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 + ((j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_0)
    t_2 = j * (exp(l) - exp(-l))
    if (l <= (-5.4d+124)) then
        tmp = t_1
    else if (l <= (-14500.0d0)) then
        tmp = t_2
    else if (l <= 0.39d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 5d+93) 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 + ((J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_0);
	double t_2 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = t_1;
	} else if (l <= -14500.0) {
		tmp = t_2;
	} else if (l <= 0.39) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 5e+93) {
		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 + ((J * (0.3333333333333333 * math.pow(l, 3.0))) * t_0)
	t_2 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if l <= -5.4e+124:
		tmp = t_1
	elif l <= -14500.0:
		tmp = t_2
	elif l <= 0.39:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 5e+93:
		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(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_0))
	t_2 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (l <= -5.4e+124)
		tmp = t_1;
	elseif (l <= -14500.0)
		tmp = t_2;
	elseif (l <= 0.39)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 5e+93)
		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 + ((J * (0.3333333333333333 * (l ^ 3.0))) * t_0);
	t_2 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if (l <= -5.4e+124)
		tmp = t_1;
	elseif (l <= -14500.0)
		tmp = t_2;
	elseif (l <= 0.39)
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 5e+93)
		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[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $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, -5.4e+124], t$95$1, If[LessEqual[l, -14500.0], t$95$2, If[LessEqual[l, 0.39], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+93], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.39999999999999956e124 or 5.0000000000000001e93 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 97.8%

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

   4. Taylor expanded in l around inf 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-*r* 97.8%

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

   6. Simplified 97.8%

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

   if -5.39999999999999956e124 < l < -14500 or 0.39000000000000001 < l < 5.0000000000000001e93

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 78.3%

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

   4. Taylor expanded in J around inf 78.3%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 99.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J (* 0.3333333333333333 (pow l 3.0))) (cos (/ K 2.0)))))
        (t_1 (* J (- (exp l) (exp (- l))))))
   (if (<= l -5.4e+124)
     t_0
     (if (<= l -14500.0)
       t_1
       (if (<= l 0.39)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (if (<= l 5e+93) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (0.3333333333333333 * pow(l, 3.0))) * cos((K / 2.0)));
	double t_1 = J * (exp(l) - exp(-l));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = t_0;
	} else if (l <= -14500.0) {
		tmp = t_1;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 5e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((j * (0.3333333333333333d0 * (l ** 3.0d0))) * cos((k / 2.0d0)))
    t_1 = j * (exp(l) - exp(-l))
    if (l <= (-5.4d+124)) then
        tmp = t_0
    else if (l <= (-14500.0d0)) then
        tmp = t_1
    else if (l <= 0.39d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 5d+93) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (0.3333333333333333 * Math.pow(l, 3.0))) * Math.cos((K / 2.0)));
	double t_1 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = t_0;
	} else if (l <= -14500.0) {
		tmp = t_1;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 5e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (0.3333333333333333 * math.pow(l, 3.0))) * math.cos((K / 2.0)))
	t_1 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if l <= -5.4e+124:
		tmp = t_0
	elif l <= -14500.0:
		tmp = t_1
	elif l <= 0.39:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 5e+93:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * cos(Float64(K / 2.0))))
	t_1 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (l <= -5.4e+124)
		tmp = t_0;
	elseif (l <= -14500.0)
		tmp = t_1;
	elseif (l <= 0.39)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 5e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (0.3333333333333333 * (l ^ 3.0))) * cos((K / 2.0)));
	t_1 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if (l <= -5.4e+124)
		tmp = t_0;
	elseif (l <= -14500.0)
		tmp = t_1;
	elseif (l <= 0.39)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 5e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.4e+124], t$95$0, If[LessEqual[l, -14500.0], t$95$1, If[LessEqual[l, 0.39], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+93], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.39999999999999956e124 or 5.0000000000000001e93 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 97.8%

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

   4. Taylor expanded in l around inf 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-*r* 97.8%

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

   6. Simplified 97.8%

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

   if -5.39999999999999956e124 < l < -14500 or 0.39000000000000001 < l < 5.0000000000000001e93

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 78.3%

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

   4. Taylor expanded in J around inf 78.3%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 98.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.11)
   (* U (+ 1.0 (* 2.0 (/ (* J (+ l (* -0.125 (* l (pow K 2.0))))) U))))
   (* U (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.11) {
		tmp = U * (1.0 + (2.0 * ((J * (l + (-0.125 * (l * pow(K, 2.0))))) / U)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / 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 (cos((k / 2.0d0)) <= (-0.11d0)) then
        tmp = u * (1.0d0 + (2.0d0 * ((j * (l + ((-0.125d0) * (l * (k ** 2.0d0))))) / u)))
    else
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    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)) <= -0.11) {
		tmp = U * (1.0 + (2.0 * ((J * (l + (-0.125 * (l * Math.pow(K, 2.0))))) / U)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.11:
		tmp = U * (1.0 + (2.0 * ((J * (l + (-0.125 * (l * math.pow(K, 2.0))))) / U)))
	else:
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.11)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0))))) / U))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.11)
		tmp = U * (1.0 + (2.0 * ((J * (l + (-0.125 * (l * (K ^ 2.0))))) / U)));
	else
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.11], N[(U * N[(1.0 + N[(2.0 * N[(N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.110000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0 51.0%

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

   4. Taylor expanded in K around 0 64.6%

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

   5. Taylor expanded in U around inf 66.2%

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

   if -0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 82.2%

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

   4. Taylor expanded in K around 0 74.3%

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

   5. Taylor expanded in U around inf 76.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.11)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (* U (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / 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 (cos((k / 2.0d0)) <= (-0.11d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    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)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.11:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.11)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.11)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.11], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.110000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0 51.0%

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

   4. Taylor expanded in K around 0 64.6%

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

   5. Taylor expanded in K around inf 64.6%

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

      1. associate-*r* 64.6%

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

      2. *-commutative 64.6%

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

   7. Simplified 64.6%

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

   if -0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 82.2%

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

   4. Taylor expanded in K around 0 74.3%

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

   5. Taylor expanded in U around inf 76.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.11)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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)) <= (-0.11d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (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)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.11:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.11)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.11)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (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], -0.11], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.110000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0 51.0%

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

   4. Taylor expanded in K around 0 64.6%

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

   5. Taylor expanded in K around inf 64.6%

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

      1. associate-*r* 64.6%

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

      2. *-commutative 64.6%

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

   7. Simplified 64.6%

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

   if -0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 82.2%

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

   4. Taylor expanded in K around 0 74.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.11)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (+ U (* J (* 0.3333333333333333 (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U + (J * (0.3333333333333333 * pow(l, 3.0)));
	}
	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)) <= (-0.11d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u + (j * (0.3333333333333333d0 * (l ** 3.0d0)))
    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)) <= -0.11) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U + (J * (0.3333333333333333 * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.11:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U + (J * (0.3333333333333333 * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.11)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.11)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U + (J * (0.3333333333333333 * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.11], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.110000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0 51.0%

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

   4. Taylor expanded in K around 0 64.6%

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

   5. Taylor expanded in K around inf 64.6%

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

      1. associate-*r* 64.6%

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

      2. *-commutative 64.6%

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

   7. Simplified 64.6%

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

   if -0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 82.2%

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

   4. Taylor expanded in K around 0 74.3%

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

   5. Taylor expanded in l around inf 65.4%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*r* 65.9%

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

   7. Simplified 65.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.11:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -14500 \lor \neg \left(\ell \leq 0.39\right):\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -14500.0) (not (<= l 0.39)))
   (* J (- (exp l) (exp (- l))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -14500.0) || !(l <= 0.39)) {
		tmp = J * (exp(l) - exp(-l));
	} 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 <= (-14500.0d0)) .or. (.not. (l <= 0.39d0))) then
        tmp = j * (exp(l) - exp(-l))
    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 <= -14500.0) || !(l <= 0.39)) {
		tmp = J * (Math.exp(l) - Math.exp(-l));
	} 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 <= -14500.0) or not (l <= 0.39):
		tmp = J * (math.exp(l) - math.exp(-l))
	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 <= -14500.0) || !(l <= 0.39))
		tmp = Float64(J * Float64(exp(l) - exp(Float64(-l))));
	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 <= -14500.0) || ~((l <= 0.39)))
		tmp = J * (exp(l) - exp(-l));
	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, -14500.0], N[Not[LessEqual[l, 0.39]], $MachinePrecision]], N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $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 < -14500 or 0.39000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in K around 0 75.0%

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

   4. Taylor expanded in J around inf 75.0%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 98.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -14500 \lor \neg \left(\ell \leq 0.39\right):\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 310:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.55e-6)
   (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
   (if (<= l 310.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (if (<= l 1.6e+37)
       (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0))))))
       (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))));
	} else if (l <= 310.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 1.6e+37) {
		tmp = 2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0)))));
	} else {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	}
	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 <= (-1.55d-6)) then
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))
    else if (l <= 310.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 1.6d+37) then
        tmp = 2.0d0 * (j * (l + ((-0.125d0) * (l * (k ** 2.0d0)))))
    else
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	} else if (l <= 310.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 1.6e+37) {
		tmp = 2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0)))));
	} else {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.55e-6:
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	elif l <= 310.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 1.6e+37:
		tmp = 2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0)))))
	else:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.55e-6)
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	elseif (l <= 310.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 1.6e+37)
		tmp = Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.55e-6)
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	elseif (l <= 310.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 1.6e+37)
		tmp = 2.0 * (J * (l + (-0.125 * (l * (K ^ 2.0)))));
	else
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.55e-6], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 310.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+37], N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.55e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 67.9%

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

   4. Taylor expanded in K around 0 51.8%

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

   5. Taylor expanded in J around inf 51.9%

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

   if -1.55e-6 < l < 310

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 99.2%

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

   if 310 < l < 1.60000000000000007e37

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 3.3%

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

   4. Taylor expanded in K around 0 60.8%

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

   5. Taylor expanded in J around inf 61.1%

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

   if 1.60000000000000007e37 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 76.0%

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

   4. Taylor expanded in K around 0 58.5%

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

   5. Taylor expanded in l around inf 58.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 310:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 320:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.55e-6)
   (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
   (if (<= l 320.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (if (<= l 2.5e+37)
       (+ U (* (* J -0.25) (* l (pow K 2.0))))
       (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))));
	} else if (l <= 320.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 2.5e+37) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	}
	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 <= (-1.55d-6)) then
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))
    else if (l <= 320.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 2.5d+37) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	} else if (l <= 320.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 2.5e+37) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.55e-6:
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	elif l <= 320.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 2.5e+37:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.55e-6)
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	elseif (l <= 320.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 2.5e+37)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.55e-6)
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	elseif (l <= 320.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 2.5e+37)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.55e-6], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 320.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e+37], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.55e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 67.9%

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

   4. Taylor expanded in K around 0 51.8%

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

   5. Taylor expanded in J around inf 51.9%

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

   if -1.55e-6 < l < 320

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 99.2%

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

   if 320 < l < 2.49999999999999994e37

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 3.3%

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

   4. Taylor expanded in K around 0 60.8%

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

   5. Taylor expanded in K around inf 60.1%

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

      1. associate-*r* 60.1%

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

      2. *-commutative 60.1%

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

   7. Simplified 60.1%

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

   if 2.49999999999999994e37 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 76.0%

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

   4. Taylor expanded in K around 0 58.5%

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

   5. Taylor expanded in l around inf 58.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 320:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6} \lor \neg \left(\ell \leq 9 \cdot 10^{-18}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.55e-6) (not (<= l 9e-18)))
   (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
   (+ U (* 2.0 (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.55e-6) || !(l <= 9e-18)) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))));
	} else {
		tmp = U + (2.0 * (J * 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 ((l <= (-1.55d-6)) .or. (.not. (l <= 9d-18))) then
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))
    else
        tmp = u + (2.0d0 * (j * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.55e-6) || !(l <= 9e-18)) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.55e-6) or not (l <= 9e-18):
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.55e-6) || !(l <= 9e-18))
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.55e-6) || ~((l <= 9e-18)))
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.55e-6], N[Not[LessEqual[l, 9e-18]], $MachinePrecision]], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.55e-6 or 8.99999999999999987e-18 < l

   1. Initial program 98.8%

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

   3. Taylor expanded in l around 0 67.3%

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

   4. Taylor expanded in K around 0 51.2%

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

   5. Taylor expanded in J around inf 51.3%

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

   if -1.55e-6 < l < 8.99999999999999987e-18

   1. Initial program 64.6%

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

   3. Taylor expanded in l around 0 99.9%

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

   4. Taylor expanded in K around 0 79.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6} \lor \neg \left(\ell \leq 9 \cdot 10^{-18}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.9 \cdot 10^{+25} \lor \neg \left(\ell \leq 4.3 \cdot 10^{+22}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.9e+25) (not (<= l 4.3e+22)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (+ U (* 2.0 (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.9e+25) || !(l <= 4.3e+22)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = U + (2.0 * (J * 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 ((l <= (-6.9d+25)) .or. (.not. (l <= 4.3d+22))) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else
        tmp = u + (2.0d0 * (j * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.9e+25) || !(l <= 4.3e+22)) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.9e+25) or not (l <= 4.3e+22):
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.9e+25) || !(l <= 4.3e+22))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.9e+25) || ~((l <= 4.3e+22)))
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.9e+25], N[Not[LessEqual[l, 4.3e+22]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.8999999999999998e25 or 4.3000000000000002e22 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 72.4%

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

   4. Taylor expanded in K around 0 55.2%

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

   5. Taylor expanded in l around inf 55.4%

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

   if -6.8999999999999998e25 < l < 4.3000000000000002e22

   1. Initial program 67.8%

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

   3. Taylor expanded in l around 0 89.5%

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

   4. Taylor expanded in K around 0 71.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.9 \cdot 10^{+25} \lor \neg \left(\ell \leq 4.3 \cdot 10^{+22}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.0% 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 84.9%

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

3. Taylor expanded in l around 0 58.7%

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

4. Taylor expanded in K around 0 46.6%

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

5. Final simplification 46.6%

   \[\leadsto U + 2 \cdot \left(J \cdot \ell\right) \]
6. Add Preprocessing

Alternative 14: 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 84.9%

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

4. Applied egg-rr 19.2%

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

5. Taylor expanded in J around 0 27.5%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.125)

C

double code(double J, double l, double K, double U) {
	return 0.125;
}

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 = 0.125d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.125;
}

Python

def code(J, l, K, U):
	return 0.125

Julia

function code(J, l, K, U)
	return 0.125
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.125;
end

Wolfram

code[J_, l_, K_, U_] := 0.125

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

5. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{0.125} \]
6. Add Preprocessing

Alternative 16: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0.037037037037037035 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.037037037037037035)

C

double code(double J, double l, double K, double U) {
	return 0.037037037037037035;
}

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 = 0.037037037037037035d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.037037037037037035;
}

Python

def code(J, l, K, U):
	return 0.037037037037037035

Julia

function code(J, l, K, U)
	return 0.037037037037037035
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.037037037037037035;
end

Wolfram

code[J_, l_, K_, U_] := 0.037037037037037035

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

5. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{0.037037037037037035} \]
6. Add Preprocessing

Alternative 17: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -8.0)

C

double code(double J, double l, double K, double U) {
	return -8.0;
}

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 = -8.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -8.0;
}

Python

def code(J, l, K, U):
	return -8.0

Julia

function code(J, l, K, U)
	return -8.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -8.0;
end

Wolfram

code[J_, l_, K_, U_] := -8.0

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

5. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-8} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))