Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 99.6%

Time: 10.0s

Alternatives: 16

Speedup: 1.4×

• 49.6% 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: 85.8% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-13)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e-13)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e-13)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e-13):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e-13))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e-13)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-13]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.0000000000000001e-13 < (-.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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-13

   1. Initial program 80.6%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+92}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -160:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 440:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)) (t_1 (cos (/ K 2.0))))
   (if (<= l -1.22e+92)
     (+ U (* t_1 (* J (* 0.3333333333333333 (pow l 3.0)))))
     (if (<= l -160.0)
       t_0
       (if (<= l 440.0)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (if (<= l 1.65e+100)
           t_0
           (+
            U
            (*
             t_1
             (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -1.22e+92) {
		tmp = U + (t_1 * (J * (0.3333333333333333 * pow(l, 3.0))));
	} else if (l <= -160.0) {
		tmp = t_0;
	} else if (l <= 440.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 1.65e+100) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    t_1 = cos((k / 2.0d0))
    if (l <= (-1.22d+92)) then
        tmp = u + (t_1 * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    else if (l <= (-160.0d0)) then
        tmp = t_0
    else if (l <= 440.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 1.65d+100) then
        tmp = t_0
    else
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -1.22e+92) {
		tmp = U + (t_1 * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	} else if (l <= -160.0) {
		tmp = t_0;
	} else if (l <= 440.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 1.65e+100) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -1.22e+92:
		tmp = U + (t_1 * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	elif l <= -160.0:
		tmp = t_0
	elif l <= 440.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 1.65e+100:
		tmp = t_0
	else:
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -1.22e+92)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))));
	elseif (l <= -160.0)
		tmp = t_0;
	elseif (l <= 440.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 1.65e+100)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -1.22e+92)
		tmp = U + (t_1 * (J * (0.3333333333333333 * (l ^ 3.0))));
	elseif (l <= -160.0)
		tmp = t_0;
	elseif (l <= 440.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 1.65e+100)
		tmp = t_0;
	else
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.22e+92], N[(U + N[(t$95$1 * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -160.0], t$95$0, If[LessEqual[l, 440.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+100], t$95$0, N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.22e92

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 97.9%

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

   4. Taylor expanded in l around inf 97.9%

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

      1. associate-*r* 97.9%

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

      2. *-commutative 97.9%

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

      3. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

   if -1.22e92 < l < -160 or 440 < l < 1.6500000000000001e100

   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 88.2%

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

   4. Taylor expanded in J around inf 88.2%

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

   if -160 < l < 440

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0 99.9%

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

   if 1.6500000000000001e100 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 97.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+92}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -160:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 440:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -260:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 150:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -1.22e+92)
     t_0
     (if (<= l -260.0)
       t_1
       (if (<= l 150.0)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (if (<= l 1.65e+100) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -1.22e+92) {
		tmp = t_0;
	} else if (l <= -260.0) {
		tmp = t_1;
	} else if (l <= 150.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 1.65e+100) {
		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 + (cos((k / 2.0d0)) * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    t_1 = (exp(l) - exp(-l)) * j
    if (l <= (-1.22d+92)) then
        tmp = t_0
    else if (l <= (-260.0d0)) then
        tmp = t_1
    else if (l <= 150.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 1.65d+100) 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.cos((K / 2.0)) * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	double t_1 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -1.22e+92) {
		tmp = t_0;
	} else if (l <= -260.0) {
		tmp = t_1;
	} else if (l <= 150.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 1.65e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	t_1 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -1.22e+92:
		tmp = t_0
	elif l <= -260.0:
		tmp = t_1
	elif l <= 150.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 1.65e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -1.22e+92)
		tmp = t_0;
	elseif (l <= -260.0)
		tmp = t_1;
	elseif (l <= 150.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 1.65e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * (l ^ 3.0))));
	t_1 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -1.22e+92)
		tmp = t_0;
	elseif (l <= -260.0)
		tmp = t_1;
	elseif (l <= 150.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 1.65e+100)
		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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1.22e+92], t$95$0, If[LessEqual[l, -260.0], t$95$1, If[LessEqual[l, 150.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.22e92 or 1.6500000000000001e100 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 97.8%

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

   4. Taylor expanded in l around inf 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-*r* 97.8%

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

   6. Simplified 97.8%

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

   if -1.22e92 < l < -260 or 150 < l < 1.6500000000000001e100

   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 88.2%

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

   4. Taylor expanded in J around inf 88.2%

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

   if -260 < l < 150

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+92}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -260:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 150:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -245 \lor \neg \left(\ell \leq 300\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -245.0) (not (<= l 300.0)))
   (* (- (exp l) (exp (- l))) J)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -245.0) || !(l <= 300.0)) {
		tmp = (exp(l) - exp(-l)) * J;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-245.0d0)) .or. (.not. (l <= 300.0d0))) then
        tmp = (exp(l) - exp(-l)) * j
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -245.0) || !(l <= 300.0)) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -245.0) or not (l <= 300.0):
		tmp = (math.exp(l) - math.exp(-l)) * J
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -245.0) || !(l <= 300.0))
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -245.0) || ~((l <= 300.0)))
		tmp = (exp(l) - exp(-l)) * J;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -245.0], N[Not[LessEqual[l, 300.0]], $MachinePrecision]], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -245 or 300 < 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 80.9%

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

   4. Taylor expanded in J around inf 80.9%

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

   if -245 < l < 300

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -245 \lor \neg \left(\ell \leq 300\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+85}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\frac{68719476736}{{K}^{3}} + -0.5}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))
   (if (<= l -3.2e+23)
     t_0
     (if (<= l 1.45e-11)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 7.8e+85)
         (* U (+ 1.0 (* J (/ (+ (/ 68719476736.0 (pow K 3.0)) -0.5) U))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	double tmp;
	if (l <= -3.2e+23) {
		tmp = t_0;
	} else if (l <= 1.45e-11) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 7.8e+85) {
		tmp = U * (1.0 + (J * (((68719476736.0 / pow(K, 3.0)) + -0.5) / U)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    if (l <= (-3.2d+23)) then
        tmp = t_0
    else if (l <= 1.45d-11) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 7.8d+85) then
        tmp = u * (1.0d0 + (j * (((68719476736.0d0 / (k ** 3.0d0)) + (-0.5d0)) / u)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	double tmp;
	if (l <= -3.2e+23) {
		tmp = t_0;
	} else if (l <= 1.45e-11) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 7.8e+85) {
		tmp = U * (1.0 + (J * (((68719476736.0 / Math.pow(K, 3.0)) + -0.5) / U)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	tmp = 0
	if l <= -3.2e+23:
		tmp = t_0
	elif l <= 1.45e-11:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 7.8e+85:
		tmp = U * (1.0 + (J * (((68719476736.0 / math.pow(K, 3.0)) + -0.5) / U)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))))
	tmp = 0.0
	if (l <= -3.2e+23)
		tmp = t_0;
	elseif (l <= 1.45e-11)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 7.8e+85)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(Float64(68719476736.0 / (K ^ 3.0)) + -0.5) / U))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	tmp = 0.0;
	if (l <= -3.2e+23)
		tmp = t_0;
	elseif (l <= 1.45e-11)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 7.8e+85)
		tmp = U * (1.0 + (J * (((68719476736.0 / (K ^ 3.0)) + -0.5) / U)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.2e+23], t$95$0, If[LessEqual[l, 1.45e-11], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+85], N[(U * N[(1.0 + N[(J * N[(N[(N[(68719476736.0 / N[Power[K, 3.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2e23 or 7.80000000000000067e85 < 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 83.0%

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

   4. Taylor expanded in K around 0 66.1%

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

   if -3.2e23 < l < 1.45e-11

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0 98.3%

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

   if 1.45e-11 < l < 7.80000000000000067e85

   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 5.2%

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

   5. Taylor expanded in K around 0 15.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 15.5%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 15.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 15.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 15.5%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 15.5%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   9. Applied egg-rr 26.3%

      \[\leadsto J \cdot \left(-0.5 + \color{blue}{{\left(K \cdot 0.000244140625\right)}^{-3}}\right) + U \]

   10. Taylor expanded in U around inf 29.7%

       \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(68719476736 \cdot \frac{1}{{K}^{3}} - 0.5\right)}{U}\right)} \]
   11. Step-by-step derivation

       1. associate-/l* 29.7%

          \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{68719476736 \cdot \frac{1}{{K}^{3}} - 0.5}{U}}\right) \]

       2. sub-neg 29.7%

          \[\leadsto U \cdot \left(1 + J \cdot \frac{\color{blue}{68719476736 \cdot \frac{1}{{K}^{3}} + \left(-0.5\right)}}{U}\right) \]

       3. associate-*r/ 29.7%

          \[\leadsto U \cdot \left(1 + J \cdot \frac{\color{blue}{\frac{68719476736 \cdot 1}{{K}^{3}}} + \left(-0.5\right)}{U}\right) \]

       4. metadata-eval 29.7%

          \[\leadsto U \cdot \left(1 + J \cdot \frac{\frac{\color{blue}{68719476736}}{{K}^{3}} + \left(-0.5\right)}{U}\right) \]

       5. metadata-eval 29.7%

          \[\leadsto U \cdot \left(1 + J \cdot \frac{\frac{68719476736}{{K}^{3}} + \color{blue}{-0.5}}{U}\right) \]

   12. Simplified 29.7%

       \[\leadsto \color{blue}{U \cdot \left(1 + J \cdot \frac{\frac{68719476736}{{K}^{3}} + -0.5}{U}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+23}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+85}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\frac{68719476736}{{K}^{3}} + -0.5}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+85}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))
   (if (<= l -7.5e+23)
     t_0
     (if (<= l 1.45e-11)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 8e+85)
         (+ U (* J (+ -0.5 (pow (* K 0.000244140625) -3.0))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	double tmp;
	if (l <= -7.5e+23) {
		tmp = t_0;
	} else if (l <= 1.45e-11) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 8e+85) {
		tmp = U + (J * (-0.5 + pow((K * 0.000244140625), -3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    if (l <= (-7.5d+23)) then
        tmp = t_0
    else if (l <= 1.45d-11) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 8d+85) then
        tmp = u + (j * ((-0.5d0) + ((k * 0.000244140625d0) ** (-3.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	double tmp;
	if (l <= -7.5e+23) {
		tmp = t_0;
	} else if (l <= 1.45e-11) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 8e+85) {
		tmp = U + (J * (-0.5 + Math.pow((K * 0.000244140625), -3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	tmp = 0
	if l <= -7.5e+23:
		tmp = t_0
	elif l <= 1.45e-11:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 8e+85:
		tmp = U + (J * (-0.5 + math.pow((K * 0.000244140625), -3.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))))
	tmp = 0.0
	if (l <= -7.5e+23)
		tmp = t_0;
	elseif (l <= 1.45e-11)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 8e+85)
		tmp = Float64(U + Float64(J * Float64(-0.5 + (Float64(K * 0.000244140625) ^ -3.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	tmp = 0.0;
	if (l <= -7.5e+23)
		tmp = t_0;
	elseif (l <= 1.45e-11)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 8e+85)
		tmp = U + (J * (-0.5 + ((K * 0.000244140625) ^ -3.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.5e+23], t$95$0, If[LessEqual[l, 1.45e-11], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+85], N[(U + N[(J * N[(-0.5 + N[Power[N[(K * 0.000244140625), $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.49999999999999987e23 or 8.0000000000000001e85 < 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 83.0%

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

   4. Taylor expanded in K around 0 66.1%

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

   if -7.49999999999999987e23 < l < 1.45e-11

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0 98.3%

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

   if 1.45e-11 < l < 8.0000000000000001e85

   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 5.2%

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

   5. Taylor expanded in K around 0 15.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 15.5%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 15.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 15.5%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 15.5%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 15.5%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   9. Applied egg-rr 26.3%

      \[\leadsto J \cdot \left(-0.5 + \color{blue}{{\left(K \cdot 0.000244140625\right)}^{-3}}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+85}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{+35}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))
   (if (<= l -1.4e+98)
     t_0
     (if (<= l -1.28e+35)
       (pow U -3.0)
       (if (<= l 1.45e-11)
         t_0
         (+ U (* J (+ -0.5 (pow (* K 0.000244140625) -3.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	double tmp;
	if (l <= -1.4e+98) {
		tmp = t_0;
	} else if (l <= -1.28e+35) {
		tmp = pow(U, -3.0);
	} else if (l <= 1.45e-11) {
		tmp = t_0;
	} else {
		tmp = U + (J * (-0.5 + pow((K * 0.000244140625), -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) :: t_0
    real(8) :: tmp
    t_0 = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    if (l <= (-1.4d+98)) then
        tmp = t_0
    else if (l <= (-1.28d+35)) then
        tmp = u ** (-3.0d0)
    else if (l <= 1.45d-11) then
        tmp = t_0
    else
        tmp = u + (j * ((-0.5d0) + ((k * 0.000244140625d0) ** (-3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	double tmp;
	if (l <= -1.4e+98) {
		tmp = t_0;
	} else if (l <= -1.28e+35) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 1.45e-11) {
		tmp = t_0;
	} else {
		tmp = U + (J * (-0.5 + Math.pow((K * 0.000244140625), -3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	tmp = 0
	if l <= -1.4e+98:
		tmp = t_0
	elif l <= -1.28e+35:
		tmp = math.pow(U, -3.0)
	elif l <= 1.45e-11:
		tmp = t_0
	else:
		tmp = U + (J * (-0.5 + math.pow((K * 0.000244140625), -3.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))))
	tmp = 0.0
	if (l <= -1.4e+98)
		tmp = t_0;
	elseif (l <= -1.28e+35)
		tmp = U ^ -3.0;
	elseif (l <= 1.45e-11)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(-0.5 + (Float64(K * 0.000244140625) ^ -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	tmp = 0.0;
	if (l <= -1.4e+98)
		tmp = t_0;
	elseif (l <= -1.28e+35)
		tmp = U ^ -3.0;
	elseif (l <= 1.45e-11)
		tmp = t_0;
	else
		tmp = U + (J * (-0.5 + ((K * 0.000244140625) ^ -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+98], t$95$0, If[LessEqual[l, -1.28e+35], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 1.45e-11], t$95$0, N[(U + N[(J * N[(-0.5 + N[Power[N[(K * 0.000244140625), $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e98 or -1.2799999999999999e35 < l < 1.45e-11

   1. Initial program 86.3%

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

   3. Taylor expanded in l around 0 78.9%

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

   if -1.4e98 < l < -1.2799999999999999e35

   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 43.9%

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

   if 1.45e-11 < 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 3.0%

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

   5. Taylor expanded in K around 0 17.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 17.3%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 17.3%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 16.1%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 16.1%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 16.1%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   9. Applied egg-rr 20.4%

      \[\leadsto J \cdot \left(-0.5 + \color{blue}{{\left(K \cdot 0.000244140625\right)}^{-3}}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{+35}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.8e+104)
   (+ U (* l (* J 2.0)))
   (if (<= l -2.7e+35)
     (pow U -3.0)
     (if (<= l 1.45e-11)
       (fma J (* l 2.0) U)
       (+ U (* J (+ -0.5 (pow (* K 0.000244140625) -3.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.8e+104) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= -2.7e+35) {
		tmp = pow(U, -3.0);
	} else if (l <= 1.45e-11) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = U + (J * (-0.5 + pow((K * 0.000244140625), -3.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.8e+104)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif (l <= -2.7e+35)
		tmp = U ^ -3.0;
	elseif (l <= 1.45e-11)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(-0.5 + (Float64(K * 0.000244140625) ^ -3.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.8e+104], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.7e+35], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 1.45e-11], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(-0.5 + N[Power[N[(K * 0.000244140625), $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8e104

   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 76.2%

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

   4. Taylor expanded in l around 0 24.1%

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

      1. associate-*r* 24.1%

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

   6. Simplified 24.1%

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

   if -1.8e104 < l < -2.70000000000000003e35

   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 43.9%

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

   if -2.70000000000000003e35 < l < 1.45e-11

   1. Initial program 81.7%

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

   3. Taylor expanded in K around 0 81.4%

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

   4. Taylor expanded in l around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. +-commutative 87.1%

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

      3. associate-*r* 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*l* 87.1%

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

      6. fma-define 87.1%

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

   6. Simplified 87.1%

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

   if 1.45e-11 < 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 3.0%

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

   5. Taylor expanded in K around 0 17.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 17.3%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 17.3%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 16.1%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 16.1%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 16.1%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   9. Applied egg-rr 20.4%

      \[\leadsto J \cdot \left(-0.5 + \color{blue}{{\left(K \cdot 0.000244140625\right)}^{-3}}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {\left(K \cdot 0.000244140625\right)}^{-3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+34}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \frac{68719476736}{{K}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.2e+103)
   (+ U (* l (* J 2.0)))
   (if (<= l -6e+34)
     (pow U -3.0)
     (if (<= l 1.45e-11)
       (fma J (* l 2.0) U)
       (+ U (* J (/ 68719476736.0 (pow K 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e+103) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= -6e+34) {
		tmp = pow(U, -3.0);
	} else if (l <= 1.45e-11) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = U + (J * (68719476736.0 / pow(K, 3.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.2e+103)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif (l <= -6e+34)
		tmp = U ^ -3.0;
	elseif (l <= 1.45e-11)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(68719476736.0 / (K ^ 3.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.2e+103], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6e+34], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 1.45e-11], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(68719476736.0 / N[Power[K, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.2000000000000003e103

   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 76.2%

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

   4. Taylor expanded in l around 0 24.1%

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

      1. associate-*r* 24.1%

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

   6. Simplified 24.1%

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

   if -4.2000000000000003e103 < l < -6.00000000000000037e34

   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 43.9%

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

   if -6.00000000000000037e34 < l < 1.45e-11

   1. Initial program 81.7%

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

   3. Taylor expanded in K around 0 81.4%

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

   4. Taylor expanded in l around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. +-commutative 87.1%

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

      3. associate-*r* 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*l* 87.1%

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

      6. fma-define 87.1%

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

   6. Simplified 87.1%

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

   if 1.45e-11 < 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 3.0%

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

   5. Taylor expanded in K around 0 17.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 17.3%

         \[\leadsto \left(\color{blue}{J \cdot -0.5} + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right) + U \]

      2. *-commutative 17.3%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{\left(J \cdot {K}^{2}\right) \cdot 0.0625}\right) + U \]

      3. associate-*l* 16.1%

         \[\leadsto \left(J \cdot -0.5 + \color{blue}{J \cdot \left({K}^{2} \cdot 0.0625\right)}\right) + U \]

      4. distribute-lft-out 16.1%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   7. Simplified 16.1%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)} + U \]

   9. Applied egg-rr 20.4%

      \[\leadsto J \cdot \left(-0.5 + \color{blue}{{\left(K \cdot 0.000244140625\right)}^{-3}}\right) + U \]

   10. Taylor expanded in K around 0 19.0%

       \[\leadsto \color{blue}{68719476736 \cdot \frac{J}{{K}^{3}}} + U \]
   11. Step-by-step derivation

       1. associate-*r/ 19.0%

          \[\leadsto \color{blue}{\frac{68719476736 \cdot J}{{K}^{3}}} + U \]

       2. *-commutative 19.0%

          \[\leadsto \frac{\color{blue}{J \cdot 68719476736}}{{K}^{3}} + U \]

       3. associate-/l* 20.3%

          \[\leadsto \color{blue}{J \cdot \frac{68719476736}{{K}^{3}}} + U \]

   12. Simplified 20.3%

       \[\leadsto \color{blue}{J \cdot \frac{68719476736}{{K}^{3}}} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+34}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \frac{68719476736}{{K}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 K around 0 80.7%

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

4. Taylor expanded in l around 0 52.5%

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

   1. associate-*r* 52.5%

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

   2. +-commutative 52.5%

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

   3. associate-*r* 52.5%

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

   4. *-commutative 52.5%

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

   5. associate-*l* 52.5%

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

   6. fma-define 52.5%

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

6. Simplified 52.5%

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

Alternative 11: 41.7% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+45} \lor \neg \left(\ell \leq 160000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.2e+45) (not (<= l 160000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e+45) || !(l <= 160000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.2d+45)) .or. (.not. (l <= 160000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e+45) || !(l <= 160000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.2e+45) or not (l <= 160000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.2e+45) || !(l <= 160000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.2e+45) || ~((l <= 160000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.2e+45], N[Not[LessEqual[l, 160000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.19999999999999995e45 or 1.6e5 < 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 15.9%

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

   if -1.19999999999999995e45 < l < 1.6e5

   1. Initial program 82.5%

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

   3. Taylor expanded in J around 0 73.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+45} \lor \neg \left(\ell \leq 160000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.7% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \mathbf{elif}\;\ell \leq 1600000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.2e+45) (* U (+ U -1.0)) (if (<= l 1600000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.2e+45) {
		tmp = U * (U + -1.0);
	} else if (l <= 1600000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.2d+45)) then
        tmp = u * (u + (-1.0d0))
    else if (l <= 1600000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.2e+45) {
		tmp = U * (U + -1.0);
	} else if (l <= 1600000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.2e+45:
		tmp = U * (U + -1.0)
	elif l <= 1600000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.2e+45)
		tmp = Float64(U * Float64(U + -1.0));
	elseif (l <= 1600000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.2e+45)
		tmp = U * (U + -1.0);
	elseif (l <= 1600000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.2e+45], N[(U * N[(U + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1600000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.19999999999999995e45

   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 76.9%

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

   5. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(U, U, -U\right)} \]
   6. Step-by-step derivation

      1. fma-undefine 18.9%

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

      2. neg-mul-1 18.9%

         \[\leadsto U \cdot U + \color{blue}{-1 \cdot U} \]

      3. distribute-rgt-out 18.9%

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

   7. Simplified 18.9%

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

   if -1.19999999999999995e45 < l < 1.6e6

   1. Initial program 82.5%

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

   3. Taylor expanded in J around 0 73.1%

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

   if 1.6e6 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 13.9%

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

Alternative 13: 53.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 K around 0 80.7%

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

4. Taylor expanded in l around 0 52.5%

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

   1. associate-*r* 52.5%

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

6. Simplified 52.5%

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

7. Final simplification 52.5%

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

Alternative 14: 36.4% 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 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 J around 0 38.6%

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

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 1.0;
}

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

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

4. Applied egg-rr 2.7%

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

   1. *-inverses 2.7%

      \[\leadsto \color{blue}{1} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Alternative 16: 2.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.0)

C

double code(double J, double l, double K, double U) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.0;
}

Python

def code(J, l, K, U):
	return 0.0

Julia

function code(J, l, K, U)
	return 0.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.0;
end

Wolfram

code[J_, l_, K_, U_] := 0.0

Derivation

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 K around 0 80.7%

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

5. Applied egg-rr 2.2%

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

   1. +-inverses 2.2%

      \[\leadsto \color{blue}{0} \]

7. Simplified 2.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))