Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.5%

Time: 9.2s

Alternatives: 10

Speedup: 1.3×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in l around 0 89.5%

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

4. Taylor expanded in l around 0 64.9%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

6. Simplified 64.9%

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

   1. log1p-expm1-u 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+25} \lor \neg \left(t_0 \leq 2 \cdot 10^{+141}\right):\\ \;\;\;\;U + t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (or (<= t_0 -2e+25) (not (<= t_0 2e+141)))
     (+ U t_0)
     (+ U (* 2.0 (* (* l (cos (* 0.5 K))) J))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if ((t_0 <= -2e+25) || !(t_0 <= 2e+141)) {
		tmp = U + t_0;
	} else {
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    if ((t_0 <= (-2d+25)) .or. (.not. (t_0 <= 2d+141))) then
        tmp = u + t_0
    else
        tmp = u + (2.0d0 * ((l * cos((0.5d0 * k))) * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if ((t_0 <= -2e+25) || !(t_0 <= 2e+141)) {
		tmp = U + t_0;
	} else {
		tmp = U + (2.0 * ((l * Math.cos((0.5 * K))) * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if (t_0 <= -2e+25) or not (t_0 <= 2e+141):
		tmp = U + t_0
	else:
		tmp = U + (2.0 * ((l * math.cos((0.5 * K))) * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if ((t_0 <= -2e+25) || !(t_0 <= 2e+141))
		tmp = Float64(U + t_0);
	else
		tmp = Float64(U + Float64(2.0 * Float64(Float64(l * cos(Float64(0.5 * K))) * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if ((t_0 <= -2e+25) || ~((t_0 <= 2e+141)))
		tmp = U + t_0;
	else
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+25], N[Not[LessEqual[t$95$0, 2e+141]], $MachinePrecision]], N[(U + t$95$0), $MachinePrecision], N[(U + N[(2.0 * N[(N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -2.00000000000000018e25 or 2.00000000000000003e141 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 74.5%

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

   if -2.00000000000000018e25 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.00000000000000003e141

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 99.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -2 \cdot 10^{+25} \lor \neg \left(J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 2 \cdot 10^{+141}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.00145:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* (pow l 3.0) (* J 0.3333333333333333)) (cos (/ K 2.0)))))
        (t_1 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -5.8e+102)
     t_0
     (if (<= l -0.00145)
       t_1
       (if (<= l 1.3e-18)
         (+ U (* 2.0 (* (* l (cos (* 0.5 K))) J)))
         (if (<= l 5.6e+102) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((pow(l, 3.0) * (J * 0.3333333333333333)) * cos((K / 2.0)));
	double t_1 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -5.8e+102) {
		tmp = t_0;
	} else if (l <= -0.00145) {
		tmp = t_1;
	} else if (l <= 1.3e-18) {
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	} else if (l <= 5.6e+102) {
		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 ** 3.0d0) * (j * 0.3333333333333333d0)) * cos((k / 2.0d0)))
    t_1 = u + (j * (exp(l) - exp(-l)))
    if (l <= (-5.8d+102)) then
        tmp = t_0
    else if (l <= (-0.00145d0)) then
        tmp = t_1
    else if (l <= 1.3d-18) then
        tmp = u + (2.0d0 * ((l * cos((0.5d0 * k))) * j))
    else if (l <= 5.6d+102) 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, 3.0) * (J * 0.3333333333333333)) * Math.cos((K / 2.0)));
	double t_1 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (l <= -5.8e+102) {
		tmp = t_0;
	} else if (l <= -0.00145) {
		tmp = t_1;
	} else if (l <= 1.3e-18) {
		tmp = U + (2.0 * ((l * Math.cos((0.5 * K))) * J));
	} else if (l <= 5.6e+102) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.pow(l, 3.0) * (J * 0.3333333333333333)) * math.cos((K / 2.0)))
	t_1 = U + (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if l <= -5.8e+102:
		tmp = t_0
	elif l <= -0.00145:
		tmp = t_1
	elif l <= 1.3e-18:
		tmp = U + (2.0 * ((l * math.cos((0.5 * K))) * J))
	elif l <= 5.6e+102:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)) * cos(Float64(K / 2.0))))
	t_1 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -5.8e+102)
		tmp = t_0;
	elseif (l <= -0.00145)
		tmp = t_1;
	elseif (l <= 1.3e-18)
		tmp = Float64(U + Float64(2.0 * Float64(Float64(l * cos(Float64(0.5 * K))) * J)));
	elseif (l <= 5.6e+102)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (((l ^ 3.0) * (J * 0.3333333333333333)) * cos((K / 2.0)));
	t_1 = U + (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (l <= -5.8e+102)
		tmp = t_0;
	elseif (l <= -0.00145)
		tmp = t_1;
	elseif (l <= 1.3e-18)
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	elseif (l <= 5.6e+102)
		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, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $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, -5.8e+102], t$95$0, If[LessEqual[l, -0.00145], t$95$1, If[LessEqual[l, 1.3e-18], N[(U + N[(2.0 * N[(N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+102], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.8000000000000005e102 or 5.60000000000000037e102 < 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 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if -5.8000000000000005e102 < l < -0.00145 or 1.3e-18 < l < 5.60000000000000037e102

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 82.9%

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

   if -0.00145 < l < 1.3e-18

   1. Initial program 71.0%

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

   3. Taylor expanded in l around 0 99.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -0.00145:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot t_0\\ t_2 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.031:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2200000:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* (* (pow l 3.0) (* J 0.3333333333333333)) t_0)))
        (t_2 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -5.8e+102)
     t_1
     (if (<= l -0.031)
       t_2
       (if (<= l 2200000.0)
         (+ U (* t_0 (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0)))))
         (if (<= l 5.6e+102) 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 + ((pow(l, 3.0) * (J * 0.3333333333333333)) * t_0);
	double t_2 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -5.8e+102) {
		tmp = t_1;
	} else if (l <= -0.031) {
		tmp = t_2;
	} else if (l <= 2200000.0) {
		tmp = U + (t_0 * (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	} else if (l <= 5.6e+102) {
		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 + (((l ** 3.0d0) * (j * 0.3333333333333333d0)) * t_0)
    t_2 = u + (j * (exp(l) - exp(-l)))
    if (l <= (-5.8d+102)) then
        tmp = t_1
    else if (l <= (-0.031d0)) then
        tmp = t_2
    else if (l <= 2200000.0d0) then
        tmp = u + (t_0 * (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (l * 2.0d0))))
    else if (l <= 5.6d+102) 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 + ((Math.pow(l, 3.0) * (J * 0.3333333333333333)) * t_0);
	double t_2 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (l <= -5.8e+102) {
		tmp = t_1;
	} else if (l <= -0.031) {
		tmp = t_2;
	} else if (l <= 2200000.0) {
		tmp = U + (t_0 * (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	} else if (l <= 5.6e+102) {
		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 + ((math.pow(l, 3.0) * (J * 0.3333333333333333)) * t_0)
	t_2 = U + (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if l <= -5.8e+102:
		tmp = t_1
	elif l <= -0.031:
		tmp = t_2
	elif l <= 2200000.0:
		tmp = U + (t_0 * (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))))
	elif l <= 5.6e+102:
		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((l ^ 3.0) * Float64(J * 0.3333333333333333)) * t_0))
	t_2 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -5.8e+102)
		tmp = t_1;
	elseif (l <= -0.031)
		tmp = t_2;
	elseif (l <= 2200000.0)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0)))));
	elseif (l <= 5.6e+102)
		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 + (((l ^ 3.0) * (J * 0.3333333333333333)) * t_0);
	t_2 = U + (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (l <= -5.8e+102)
		tmp = t_1;
	elseif (l <= -0.031)
		tmp = t_2;
	elseif (l <= 2200000.0)
		tmp = U + (t_0 * (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0))));
	elseif (l <= 5.6e+102)
		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[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.8e+102], t$95$1, If[LessEqual[l, -0.031], t$95$2, If[LessEqual[l, 2200000.0], N[(U + N[(t$95$0 * N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+102], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.8000000000000005e102 or 5.60000000000000037e102 < 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 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if -5.8000000000000005e102 < l < -0.031 or 2.2e6 < l < 5.60000000000000037e102

   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 85.3%

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

   if -0.031 < l < 2.2e6

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 98.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -0.031:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2200000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.045:\\ \;\;\;\;U + -0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.045)
   (+ U (* -0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.045) {
		tmp = U + (-0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.045d0)) then
        tmp = u + ((-0.3333333333333333d0) * (j * (l ** 3.0d0)))
    else
        tmp = u + (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.045) {
		tmp = U + (-0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.045:
		tmp = U + (-0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.045)
		tmp = Float64(U + Float64(-0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.045)
		tmp = U + (-0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.045], N[(U + N[(-0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.044999999999999998

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0 92.2%

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

   4. Taylor expanded in K around 0 33.7%

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

      1. add-sqr-sqrt 30.5%

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

      2. sqrt-unprod 57.6%

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

      3. *-commutative 57.6%

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

      4. *-commutative 57.6%

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

      5. swap-sqr 57.6%

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

      6. pow-prod-up 57.6%

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

      7. metadata-eval 57.6%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{\color{blue}{6}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} + 2 \cdot \ell\right) + U \]

      8. metadata-eval 57.6%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{6} \cdot \color{blue}{0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   6. Applied egg-rr 57.6%

      \[\leadsto J \cdot \left(\color{blue}{\sqrt{{\ell}^{6} \cdot 0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   7. Taylor expanded in l around -inf 75.8%

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

      1. *-commutative 75.8%

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

   9. Simplified 75.8%

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

   if -0.044999999999999998 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   3. Taylor expanded in l around 0 88.6%

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

   4. Taylor expanded in K around 0 85.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.045:\\ \;\;\;\;U + -0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot {\ell}^{3}\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.045:\\ \;\;\;\;U + -0.3333333333333333 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (pow l 3.0))))
   (if (<= (cos (/ K 2.0)) -0.045)
     (+ U (* -0.3333333333333333 t_0))
     (+ U (* 0.3333333333333333 t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * pow(l, 3.0);
	double tmp;
	if (cos((K / 2.0)) <= -0.045) {
		tmp = U + (-0.3333333333333333 * t_0);
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (l ** 3.0d0)
    if (cos((k / 2.0d0)) <= (-0.045d0)) then
        tmp = u + ((-0.3333333333333333d0) * t_0)
    else
        tmp = u + (0.3333333333333333d0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * Math.pow(l, 3.0);
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.045) {
		tmp = U + (-0.3333333333333333 * t_0);
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * math.pow(l, 3.0)
	tmp = 0
	if math.cos((K / 2.0)) <= -0.045:
		tmp = U + (-0.3333333333333333 * t_0)
	else:
		tmp = U + (0.3333333333333333 * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * (l ^ 3.0))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.045)
		tmp = Float64(U + Float64(-0.3333333333333333 * t_0));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l ^ 3.0);
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.045)
		tmp = U + (-0.3333333333333333 * t_0);
	else
		tmp = U + (0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.045], N[(U + N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.044999999999999998

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0 92.2%

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

   4. Taylor expanded in K around 0 33.7%

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

      1. add-sqr-sqrt 30.5%

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

      2. sqrt-unprod 57.6%

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

      3. *-commutative 57.6%

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

      4. *-commutative 57.6%

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

      5. swap-sqr 57.6%

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

      6. pow-prod-up 57.6%

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

      7. metadata-eval 57.6%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{\color{blue}{6}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} + 2 \cdot \ell\right) + U \]

      8. metadata-eval 57.6%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{6} \cdot \color{blue}{0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   6. Applied egg-rr 57.6%

      \[\leadsto J \cdot \left(\color{blue}{\sqrt{{\ell}^{6} \cdot 0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   7. Taylor expanded in l around -inf 75.8%

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

      1. *-commutative 75.8%

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

   9. Simplified 75.8%

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

   if -0.044999999999999998 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   3. Taylor expanded in l around 0 88.6%

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

   4. Taylor expanded in K around 0 85.3%

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

   5. Taylor expanded in l around inf 74.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.045:\\ \;\;\;\;U + -0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -780000 \lor \neg \left(\ell \leq 4.4\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -780000.0) (not (<= l 4.4)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* 2.0 (* (* l (cos (* 0.5 K))) J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -780000.0) || !(l <= 4.4)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-780000.0d0)) .or. (.not. (l <= 4.4d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (2.0d0 * ((l * cos((0.5d0 * k))) * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -780000.0) || !(l <= 4.4)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * ((l * Math.cos((0.5 * K))) * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -780000.0) or not (l <= 4.4):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (2.0 * ((l * math.cos((0.5 * K))) * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -780000.0) || !(l <= 4.4))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(Float64(l * cos(Float64(0.5 * K))) * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -780000.0) || ~((l <= 4.4)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (2.0 * ((l * cos((0.5 * K))) * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -780000.0], N[Not[LessEqual[l, 4.4]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.8e5 or 4.4000000000000004 < 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 79.1%

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

   4. Taylor expanded in K around 0 58.7%

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

   5. Taylor expanded in l around inf 58.7%

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

   if -7.8e5 < l < 4.4000000000000004

   1. Initial program 71.9%

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

   3. Taylor expanded in l around 0 99.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -780000 \lor \neg \left(\ell \leq 4.4\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 170000000000:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + -0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 170000000000.0)
   (+ U (* J (* l 2.0)))
   (+ U (* -0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 170000000000.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (-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 (k <= 170000000000.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = u + ((-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 (K <= 170000000000.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (-0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 170000000000.0:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = U + (-0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 170000000000.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(U + 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 (K <= 170000000000.0)
		tmp = U + (J * (l * 2.0));
	else
		tmp = U + (-0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 170000000000.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(-0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.7e11

   1. Initial program 87.3%

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

   3. Taylor expanded in l around 0 63.9%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*l* 63.8%

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

      4. associate-*r* 63.8%

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

   5. Simplified 63.8%

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

   6. Taylor expanded in K around 0 57.6%

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

   if 1.7e11 < K

   1. Initial program 80.5%

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

   3. Taylor expanded in l around 0 90.5%

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

   4. Taylor expanded in K around 0 57.4%

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

      1. add-sqr-sqrt 39.2%

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

      2. sqrt-unprod 55.5%

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

      3. *-commutative 55.5%

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

      4. *-commutative 55.5%

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

      5. swap-sqr 55.5%

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

      6. pow-prod-up 55.5%

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

      7. metadata-eval 55.5%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{\color{blue}{6}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} + 2 \cdot \ell\right) + U \]

      8. metadata-eval 55.5%

         \[\leadsto J \cdot \left(\sqrt{{\ell}^{6} \cdot \color{blue}{0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   6. Applied egg-rr 55.5%

      \[\leadsto J \cdot \left(\color{blue}{\sqrt{{\ell}^{6} \cdot 0.1111111111111111}} + 2 \cdot \ell\right) + U \]

   7. Taylor expanded in l around -inf 52.3%

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

      1. *-commutative 52.3%

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

   9. Simplified 52.3%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 170000000000:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + -0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in l around 0 64.9%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 64.9%

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

   4. associate-*r* 64.9%

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

5. Simplified 64.9%

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

6. Taylor expanded in K around 0 55.4%

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

7. Final simplification 55.4%

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

Alternative 10: 36.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in l around 0 89.5%

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

4. Taylor expanded in l around inf 75.1%

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

   1. *-commutative 75.1%

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

   2. *-commutative 75.1%

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

   3. associate-*l* 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in l around 0 36.3%

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

8. Final simplification 36.3%

   \[\leadsto U \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))