Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.4%

Time: 12.0s

Alternatives: 11

Speedup: 1.5×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% 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 82.4%

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

2. Taylor expanded in l around 0 64.6%

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

   1. log1p-expm1-u 99.1%

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

   2. *-commutative 99.1%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.00068:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.00055:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* (pow l 5.0) (* J 0.016666666666666666)) (cos (/ K 2.0)))))
        (t_1 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -4.2e+90)
     t_0
     (if (<= l -0.00068)
       t_1
       (if (<= l 0.00055)
         (+ U (* (* J l) (* 2.0 (cos (* K 0.5)))))
         (if (<= l 4.5e+61) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((pow(l, 5.0) * (J * 0.016666666666666666)) * cos((K / 2.0)));
	double t_1 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -4.2e+90) {
		tmp = t_0;
	} else if (l <= -0.00068) {
		tmp = t_1;
	} else if (l <= 0.00055) {
		tmp = U + ((J * l) * (2.0 * cos((K * 0.5))));
	} else if (l <= 4.5e+61) {
		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 + (((l ** 5.0d0) * (j * 0.016666666666666666d0)) * cos((k / 2.0d0)))
    t_1 = u + (j * (exp(l) - exp(-l)))
    if (l <= (-4.2d+90)) then
        tmp = t_0
    else if (l <= (-0.00068d0)) then
        tmp = t_1
    else if (l <= 0.00055d0) then
        tmp = u + ((j * l) * (2.0d0 * cos((k * 0.5d0))))
    else if (l <= 4.5d+61) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((Math.pow(l, 5.0) * (J * 0.016666666666666666)) * Math.cos((K / 2.0)));
	double t_1 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (l <= -4.2e+90) {
		tmp = t_0;
	} else if (l <= -0.00068) {
		tmp = t_1;
	} else if (l <= 0.00055) {
		tmp = U + ((J * l) * (2.0 * Math.cos((K * 0.5))));
	} else if (l <= 4.5e+61) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.pow(l, 5.0) * (J * 0.016666666666666666)) * math.cos((K / 2.0)))
	t_1 = U + (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if l <= -4.2e+90:
		tmp = t_0
	elif l <= -0.00068:
		tmp = t_1
	elif l <= 0.00055:
		tmp = U + ((J * l) * (2.0 * math.cos((K * 0.5))))
	elif l <= 4.5e+61:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64((l ^ 5.0) * Float64(J * 0.016666666666666666)) * cos(Float64(K / 2.0))))
	t_1 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -4.2e+90)
		tmp = t_0;
	elseif (l <= -0.00068)
		tmp = t_1;
	elseif (l <= 0.00055)
		tmp = Float64(U + Float64(Float64(J * l) * Float64(2.0 * cos(Float64(K * 0.5)))));
	elseif (l <= 4.5e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (((l ^ 5.0) * (J * 0.016666666666666666)) * cos((K / 2.0)));
	t_1 = U + (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (l <= -4.2e+90)
		tmp = t_0;
	elseif (l <= -0.00068)
		tmp = t_1;
	elseif (l <= 0.00055)
		tmp = U + ((J * l) * (2.0 * cos((K * 0.5))));
	elseif (l <= 4.5e+61)
		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[(N[Power[l, 5.0], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2e+90], t$95$0, If[LessEqual[l, -0.00068], t$95$1, If[LessEqual[l, 0.00055], N[(U + N[(N[(J * l), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+61], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.19999999999999961e90 or 4.5e61 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   5. Simplified 100.0%

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

   if -4.19999999999999961e90 < l < -6.8e-4 or 5.50000000000000033e-4 < l < 4.5e61

   1. Initial program 99.7%

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

   2. Taylor expanded in K around 0 87.2%

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

   if -6.8e-4 < l < 5.50000000000000033e-4

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;U + \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -0.00068:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 0.00055:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left({\ell}^{5} \cdot \left(J \cdot 0.016666666666666666\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.08], N[(U + N[(N[(J * l), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.0%

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

   2. Taylor expanded in l around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*r* 74.6%

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

      3. associate-*l* 74.6%

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

      4. *-commutative 74.6%

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

   4. Simplified 74.6%

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

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

   1. Initial program 83.4%

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

   2. Taylor expanded in l around 0 88.2%

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

   3. Taylor expanded in K around 0 84.9%

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

4. Final simplification 82.6%

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

Alternative 4: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00015 \lor \neg \left(\ell \leq 2.8 \cdot 10^{-5}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00015) (not (<= l 2.8e-5)))
   (+ U (* J (- (exp l) (exp (- l)))))
   (+ U (* (* J l) (* 2.0 (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00015) || !(l <= 2.8e-5)) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = U + ((J * l) * (2.0 * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.00015d0)) .or. (.not. (l <= 2.8d-5))) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = u + ((j * l) * (2.0d0 * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00015) || !(l <= 2.8e-5)) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = U + ((J * l) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.00015) or not (l <= 2.8e-5):
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = U + ((J * l) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00015) || !(l <= 2.8e-5))
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(Float64(J * l) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.00015) || ~((l <= 2.8e-5)))
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = U + ((J * l) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00015], N[Not[LessEqual[l, 2.8e-5]], $MachinePrecision]], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * l), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.49999999999999987e-4 or 2.79999999999999996e-5 < l

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 80.9%

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

   if -1.49999999999999987e-4 < l < 2.79999999999999996e-5

   1. Initial program 65.7%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00015 \lor \neg \left(\ell \leq 2.8 \cdot 10^{-5}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 63.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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 * cos((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
end

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in l around 0 64.6%

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

3. Final simplification 64.6%

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

Alternative 6: 64.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + ((j * l) * (2.0d0 * cos((k * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + ((J * l) * (2.0 * cos((K * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in l around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. associate-*r* 64.6%

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

   3. associate-*l* 64.6%

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

   4. *-commutative 64.6%

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

4. Simplified 64.6%

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

5. Final simplification 64.6%

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

Alternative 7: 42.4% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -750:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.45:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -750.0) (* U U) (if (<= l 1.45) U (* U (- U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -750.0) {
		tmp = U * U;
	} else if (l <= 1.45) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	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 <= (-750.0d0)) then
        tmp = u * u
    else if (l <= 1.45d0) then
        tmp = u
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -750.0) {
		tmp = U * U;
	} else if (l <= 1.45) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -750.0:
		tmp = U * U
	elif l <= 1.45:
		tmp = U
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -750.0)
		tmp = Float64(U * U);
	elseif (l <= 1.45)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -750.0)
		tmp = U * U;
	elseif (l <= 1.45)
		tmp = U;
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -750.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.45], U, N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -750

   1. Initial program 100.0%

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

   3. Applied egg-rr 23.9%

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

   if -750 < l < 1.44999999999999996

   1. Initial program 66.7%

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

   2. Taylor expanded in J around 0 64.7%

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

   if 1.44999999999999996 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 10.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -750:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.45:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]

Alternative 8: 42.2% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -660:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{+52}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-4 - U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -660.0) (* U U) (if (<= l 1.76e+52) U (- -4.0 (* U U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -660.0) {
		tmp = U * U;
	} else if (l <= 1.76e+52) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-660.0d0)) then
        tmp = u * u
    else if (l <= 1.76d+52) then
        tmp = u
    else
        tmp = (-4.0d0) - (u * u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -660.0) {
		tmp = U * U;
	} else if (l <= 1.76e+52) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -660.0:
		tmp = U * U
	elif l <= 1.76e+52:
		tmp = U
	else:
		tmp = -4.0 - (U * U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -660.0)
		tmp = Float64(U * U);
	elseif (l <= 1.76e+52)
		tmp = U;
	else
		tmp = Float64(-4.0 - Float64(U * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -660.0)
		tmp = U * U;
	elseif (l <= 1.76e+52)
		tmp = U;
	else
		tmp = -4.0 - (U * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -660.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.76e+52], U, N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -660

   1. Initial program 100.0%

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

   3. Applied egg-rr 23.9%

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

   if -660 < l < 1.75999999999999993e52

   1. Initial program 68.5%

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

   2. Taylor expanded in J around 0 61.2%

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

   if 1.75999999999999993e52 < l

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 15.4%

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

      1. cancel-sign-sub-inv 15.4%

         \[\leadsto \color{blue}{-4 - U \cdot U} \]

   7. Simplified 15.4%

      \[\leadsto \color{blue}{-4 - U \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -660:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{+52}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-4 - U \cdot U\\ \end{array} \]

Alternative 9: 42.0% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -22000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{+63}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -22000.0) (not (<= l 6.2e+63))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -22000.0) || !(l <= 6.2e+63)) {
		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 <= (-22000.0d0)) .or. (.not. (l <= 6.2d+63))) 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 <= -22000.0) || !(l <= 6.2e+63)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -22000.0) or not (l <= 6.2e+63):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -22000.0) || !(l <= 6.2e+63))
		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 <= -22000.0) || ~((l <= 6.2e+63)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -22000.0], N[Not[LessEqual[l, 6.2e+63]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -22000 or 6.2000000000000001e63 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 18.6%

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

   if -22000 < l < 6.2000000000000001e63

   1. Initial program 68.9%

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

   2. Taylor expanded in J around 0 60.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -22000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{+63}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 10: 54.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in l around 0 64.6%

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

3. Taylor expanded in K around 0 53.9%

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

   1. +-commutative 53.9%

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

   2. associate-*r* 53.9%

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

   3. *-commutative 53.9%

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

5. Simplified 53.9%

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

6. Final simplification 53.9%

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

Alternative 11: 36.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in J around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto U \]

Reproduce

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