Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 97.6%

Time: 11.6s

Alternatives: 15

Speedup: 2.7×

• 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: 86.8% 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: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -0.74:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(0.3333333333333333 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 0.0039:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -0.74)
     (+ U (* t_1 (* J (- 0.3333333333333333 t_0))))
     (if (<= l 0.0039)
       (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))
       (if (<= l 8e+102)
         (+ U (* J (- (exp l) t_0)))
         (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -0.74) {
		tmp = U + (t_1 * (J * (0.3333333333333333 - t_0)));
	} else if (l <= 0.0039) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else if (l <= 8e+102) {
		tmp = U + (J * (exp(l) - t_0));
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-0.74d0)) then
        tmp = u + (t_1 * (j * (0.3333333333333333d0 - t_0)))
    else if (l <= 0.0039d0) then
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    else if (l <= 8d+102) then
        tmp = u + (j * (exp(l) - t_0))
    else
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -0.74) {
		tmp = U + (t_1 * (J * (0.3333333333333333 - t_0)));
	} else if (l <= 0.0039) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else if (l <= 8e+102) {
		tmp = U + (J * (Math.exp(l) - t_0));
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -0.74:
		tmp = U + (t_1 * (J * (0.3333333333333333 - t_0)))
	elif l <= 0.0039:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	elif l <= 8e+102:
		tmp = U + (J * (math.exp(l) - t_0))
	else:
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -0.74)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(0.3333333333333333 - t_0))));
	elseif (l <= 0.0039)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 8e+102)
		tmp = Float64(U + Float64(J * Float64(exp(l) - t_0)));
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -0.74)
		tmp = U + (t_1 * (J * (0.3333333333333333 - t_0)));
	elseif (l <= 0.0039)
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	elseif (l <= 8e+102)
		tmp = U + (J * (exp(l) - t_0));
	else
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -0.74], N[(U + N[(t$95$1 * N[(J * N[(0.3333333333333333 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.0039], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+102], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -0.73999999999999999

   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

   4. Applied egg-rr 98.5%

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

   if -0.73999999999999999 < l < 0.0038999999999999998

   1. Initial program 75.9%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-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-*l* 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if 0.0038999999999999998 < l < 7.99999999999999982e102

   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 85.0%

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

   if 7.99999999999999982e102 < 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%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.74:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 0.0039:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in l around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*l* 60.5%

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

5. Simplified 60.5%

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

   1. log1p-expm1-u 98.2%

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

7. Applied egg-rr 98.2%

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

Alternative 3: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.325)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) 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)) <= 0.325) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / 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)) <= 0.325d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / 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)) <= 0.325) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / 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)) <= 0.325:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / 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)) <= 0.325)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / 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)) <= 0.325)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / 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], 0.325], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $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))) < 0.325000000000000011

   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 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-expm1-u 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in U around inf 66.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 75.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   10. Simplified 75.1%

       \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   if 0.325000000000000011 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in l around 0 85.8%

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

   4. Taylor expanded in K around 0 83.2%

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

   5. Taylor expanded in U around inf 83.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.325)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.325) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * 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)) <= 0.325d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * 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)) <= 0.325) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.325:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.325)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.325)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.325], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.325000000000000011

   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 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-expm1-u 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in U around inf 66.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 75.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   10. Simplified 75.1%

       \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   if 0.325000000000000011 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in l around 0 85.8%

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

   4. Taylor expanded in K around 0 83.2%

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

      1. unpow2 85.8%

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

   6. Applied egg-rr 83.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.325)
   (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.325) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * 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)) <= 0.325d0) then
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * 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)) <= 0.325) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.325:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.325)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.325)
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.325], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.325000000000000011

   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 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 62.1%

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

      3. *-commutative 62.1%

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

      4. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if 0.325000000000000011 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in l around 0 85.8%

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

   4. Taylor expanded in K around 0 83.2%

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

      1. unpow2 85.8%

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

   6. Applied egg-rr 83.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.325:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-132)
   (+ U (* J (- (exp l) (exp (- l)))))
   (+
    U
    (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-132) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * 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 ((k / 2.0d0) <= 5d-132) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-132) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-132:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-132)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-132)
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-132], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.9999999999999999e-132

   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 K around 0 70.5%

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

   if 4.9999999999999999e-132 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 91.0%

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

   3. Taylor expanded in l around 0 92.8%

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

      1. unpow2 92.8%

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

   5. Applied egg-rr 92.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * 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 + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
end

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in l around 0 89.2%

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

   1. unpow2 89.2%

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

5. Applied egg-rr 89.2%

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

6. Final simplification 89.2%

   \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 8: 59.4% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 10500.0)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* (* J l) -6.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + ((J * l) * -6.0);
	}
	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 <= 10500.0d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + ((j * l) * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + ((J * l) * -6.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 10500.0:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + ((J * l) * -6.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 10500.0)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(Float64(J * l) * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 10500.0)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + ((J * l) * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 10500.0], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * l), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 10500

   1. Initial program 87.5%

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

   3. Taylor expanded in l around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

   5. Simplified 61.8%

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

      1. log1p-expm1-u 97.6%

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

   7. Applied egg-rr 97.6%

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

   8. Taylor expanded in K around 0 49.2%

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

      1. associate-*r* 49.2%

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

   10. Simplified 49.2%

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

   11. Taylor expanded in U around inf 53.3%

       \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 55.3%

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

   13. Simplified 55.3%

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

   if 10500 < K

   1. Initial program 92.6%

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

   3. Taylor expanded in l around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

   5. Simplified 55.9%

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

   7. Applied egg-rr 46.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]

   8. Taylor expanded in K around 0 44.5%

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

      1. *-commutative 44.5%

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

   10. Simplified 44.5%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.7% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \lor \neg \left(\ell \leq 120\right):\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.5) (not (<= l 120.0))) (/ (* U U) U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.5) || !(l <= 120.0)) {
		tmp = (U * 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 <= (-5.5d0)) .or. (.not. (l <= 120.0d0))) then
        tmp = (u * 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 <= -5.5) || !(l <= 120.0)) {
		tmp = (U * U) / U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.5) or not (l <= 120.0):
		tmp = (U * U) / U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.5) || !(l <= 120.0))
		tmp = Float64(Float64(U * U) / U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.5) || ~((l <= 120.0)))
		tmp = (U * U) / U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.5], N[Not[LessEqual[l, 120.0]], $MachinePrecision]], N[(N[(U * U), $MachinePrecision] / U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.5 or 120 < 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

   4. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{0} + U \]
   5. Step-by-step derivation

      1. flip-+ 20.6%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - U \cdot U}{0 - U}} \]

      2. metadata-eval 20.6%

         \[\leadsto \frac{\color{blue}{0} - U \cdot U}{0 - U} \]

      3. div-sub 20.6%

         \[\leadsto \color{blue}{\frac{0}{0 - U} - \frac{U \cdot U}{0 - U}} \]

      4. sub0-neg 20.6%

         \[\leadsto \frac{0}{\color{blue}{-U}} - \frac{U \cdot U}{0 - U} \]

      5. pow2 20.6%

         \[\leadsto \frac{0}{-U} - \frac{\color{blue}{{U}^{2}}}{0 - U} \]

      6. sub0-neg 20.6%

         \[\leadsto \frac{0}{-U} - \frac{{U}^{2}}{\color{blue}{-U}} \]

   6. Applied egg-rr 20.6%

      \[\leadsto \color{blue}{\frac{0}{-U} - \frac{{U}^{2}}{-U}} \]
   7. Step-by-step derivation

      1. div0 20.6%

         \[\leadsto \color{blue}{0} - \frac{{U}^{2}}{-U} \]

      2. neg-sub0 20.6%

         \[\leadsto \color{blue}{-\frac{{U}^{2}}{-U}} \]

      3. distribute-frac-neg2 20.6%

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

      4. remove-double-neg 20.6%

         \[\leadsto \frac{{U}^{2}}{\color{blue}{U}} \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{{U}^{2}}{U}} \]
   9. Step-by-step derivation

      1. unpow2 20.6%

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

   10. Applied egg-rr 20.6%

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

   if -5.5 < l < 120

   1. Initial program 76.3%

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

   4. Applied egg-rr 57.1%

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

   5. Taylor expanded in U around inf 74.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \lor \neg \left(\ell \leq 120\right):\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * 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 + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in l around 0 89.2%

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

4. Taylor expanded in K around 0 66.3%

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

   1. unpow2 89.2%

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

6. Applied egg-rr 66.3%

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

7. Final simplification 66.3%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \]
8. Add Preprocessing

Alternative 11: 54.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 10500.0) (+ U (* 2.0 (* J l))) (+ U (* (* J l) -6.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U + (2.0 * (J * l));
	} else {
		tmp = U + ((J * l) * -6.0);
	}
	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 <= 10500.0d0) then
        tmp = u + (2.0d0 * (j * l))
    else
        tmp = u + ((j * l) * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U + (2.0 * (J * l));
	} else {
		tmp = U + ((J * l) * -6.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 10500.0:
		tmp = U + (2.0 * (J * l))
	else:
		tmp = U + ((J * l) * -6.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 10500.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	else
		tmp = Float64(U + Float64(Float64(J * l) * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 10500.0)
		tmp = U + (2.0 * (J * l));
	else
		tmp = U + ((J * l) * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 10500.0], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * l), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 10500

   1. Initial program 87.5%

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

   3. Taylor expanded in l around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in K around 0 49.2%

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

   if 10500 < K

   1. Initial program 92.6%

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

   3. Taylor expanded in l around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

   5. Simplified 55.9%

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

   7. Applied egg-rr 46.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]

   8. Taylor expanded in K around 0 44.5%

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

      1. *-commutative 44.5%

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

   10. Simplified 44.5%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \ell\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 10500.0) (+ U (* 2.0 (* J l))) (+ U (* J (* l -6.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U + (2.0 * (J * l));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	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 <= 10500.0d0) then
        tmp = u + (2.0d0 * (j * l))
    else
        tmp = u + (j * (l * (-6.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 10500.0) {
		tmp = U + (2.0 * (J * l));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 10500.0:
		tmp = U + (2.0 * (J * l))
	else:
		tmp = U + (J * (l * -6.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 10500.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	else
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 10500.0)
		tmp = U + (2.0 * (J * l));
	else
		tmp = U + (J * (l * -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 10500.0], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 10500

   1. Initial program 87.5%

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

   3. Taylor expanded in l around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in K around 0 49.2%

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

   if 10500 < K

   1. Initial program 92.6%

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

   3. Taylor expanded in l around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

   5. Simplified 55.9%

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

   7. Applied egg-rr 46.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]

   8. Taylor expanded in K around 0 44.5%

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

      1. *-commutative 44.5%

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

      2. associate-*r* 44.5%

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   10. Simplified 44.5%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 10500:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+51}:\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.16e+51) (/ (* U U) U) (+ U (* 2.0 (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.16e+51) {
		tmp = (U * U) / U;
	} 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 <= (-1.16d+51)) then
        tmp = (u * u) / u
    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 <= -1.16e+51) {
		tmp = (U * U) / U;
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.16e+51:
		tmp = (U * U) / U
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.16e+51)
		tmp = Float64(Float64(U * U) / U);
	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 <= -1.16e+51)
		tmp = (U * U) / U;
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.16e+51], N[(N[(U * U), $MachinePrecision] / U), $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.16e51

   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

   4. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{0} + U \]
   5. Step-by-step derivation

      1. flip-+ 25.1%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - U \cdot U}{0 - U}} \]

      2. metadata-eval 25.1%

         \[\leadsto \frac{\color{blue}{0} - U \cdot U}{0 - U} \]

      3. div-sub 25.1%

         \[\leadsto \color{blue}{\frac{0}{0 - U} - \frac{U \cdot U}{0 - U}} \]

      4. sub0-neg 25.1%

         \[\leadsto \frac{0}{\color{blue}{-U}} - \frac{U \cdot U}{0 - U} \]

      5. pow2 25.1%

         \[\leadsto \frac{0}{-U} - \frac{\color{blue}{{U}^{2}}}{0 - U} \]

      6. sub0-neg 25.1%

         \[\leadsto \frac{0}{-U} - \frac{{U}^{2}}{\color{blue}{-U}} \]

   6. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{\frac{0}{-U} - \frac{{U}^{2}}{-U}} \]
   7. Step-by-step derivation

      1. div0 25.1%

         \[\leadsto \color{blue}{0} - \frac{{U}^{2}}{-U} \]

      2. neg-sub0 25.1%

         \[\leadsto \color{blue}{-\frac{{U}^{2}}{-U}} \]

      3. distribute-frac-neg2 25.1%

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

      4. remove-double-neg 25.1%

         \[\leadsto \frac{{U}^{2}}{\color{blue}{U}} \]

   8. Simplified 25.1%

      \[\leadsto \color{blue}{\frac{{U}^{2}}{U}} \]
   9. Step-by-step derivation

      1. unpow2 25.1%

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

   10. Applied egg-rr 25.1%

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

   if -1.16e51 < l

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*l* 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in K around 0 57.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+51}:\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.2% 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 88.7%

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

4. Applied egg-rr 28.9%

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

5. Taylor expanded in U around inf 37.5%

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

Alternative 15: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

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 = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 88.7%

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

4. Applied egg-rr 28.9%

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

5. Taylor expanded in U around 0 3.1%

   \[\leadsto \color{blue}{-4} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))